Cinelerra | Open Source Video Editing

"Cinelerra is a professional video editing and compositing software. It is designed for the GNU/Linux operating system. It is produced by Heroine Virtual, and is free software distributed under the GNU General Public License. Cinelerra also includes a video compositing engine, allowing the user to perform advanced compositing operations such as keying and mattes."

http://cinelerra.org/

GimpFX Foundry | free scripts to expand GIMP

"This set contains different scripts for the creation of effects such as the old picture, lomography, image stylization, creation of all possible meshes and frames for pictures. It also contains scripts that significantly extend GIMP options when working with layers.
The set includes a huge number of scripts for work with image color; there are the scripts for total picture editing, creation of inner shadow, picture sphere and many other effects." [1]
(Link)

Geeqie | free image management software

"Geeqie is an image viewer that makes navigation of images easy. Images can be viewed full screen, and a slide show function allows automatic changing of the displayed image.
Geeqie is an image manager, with complete copy, move, rename, and delete functions available at the press of a button. Find the right image with the search feature. Image collectors can free up space with a function that finds duplicate images.
The one thing Geeqie is not, and never will be, is an image editor. However, Geeqie paired with an image editor, such as the Gimp, can be a very powerful combination."

(Link)

Arduino | open source programming and micro-controller

"Arduino is an open-source electronics prototyping platform based on flexible, easy-to-use hardware and software. It's intended for artists, designers, hobbyists and anyone interested in creating interactive objects or environments. Arduino can sense the environment by receiving input from a variety of sensors and can affect its surroundings by controlling lights, motors, and other actuators. The microcontroller on the board is programmed using the Arduino programming language (based on Wiring) and the Arduino development environment (based on Processing). Arduino projects can be stand-alone or they can communicate with software running on a computer (e.g. Flash, Processing, MaxMSP)."

(Link)

Longomatch | open source sports video analysis

"LongoMatch has been designed to be very easy to use, exposing the basic functionalities of video analysis in an intuitive interface. Tagging, playback and edition of stored events can be easily done from the main window, while more specific features can be accessed through menus when needed. Flexible and customizable for all kind of sports Flexible and customizable for all sports LongoMatch can be used for any kind of sports, allowing to create custom templates with an unlimited number of tagging categories. It also supports defining custom subcategories and creating templates for your teams with detailed information of each player which is the perfect combination for a fine-grained performance analysis"
(Link)

Pencil | open source animation and drawing software

Update:
@ibeardslee has pointed me to a forked, updated version of this software called Pencil2D, which appears to be a better option than the link I give below.
(Link)
"Pencil is an animation/drawing software for Mac OS X, Windows, and Linux. It lets you create traditional hand-drawn animation (cartoon) using both bitmap and vector graphics. Pencil is free and open source.

Pencil is not intended to mimick web-oriented vector-based animation software such as Flash. Its main purpose is to make traditional animation. Neither does it try to rival commercial software targetting the professional animation sector. Pencil is intended to be a simple programme enabling anyone to make 2D animation."
(Link)

Drupal | open source community & content management platform

Drupal is an open source content management platform powering millions of websites and applications. It’s built, used, and supported by an active and diverse community of people around the world.
Use Drupal to build everything from personal blogs to enterprise applications. Thousands of add-on modules and designs let you build any site you can imagine.
(Link)
Drupal's Video Module (with a few other tweaks) turns Drupal into something approaching Youtube as a video sharing site.

Clementine | free music and audio file player

"Clementine is a modern music player and library organizer. It's a multiplatform music player, inspired by Amarok 1.4, focusing on a fast and easy-to-use interface for searching and playing your music.

Features:

• Search and play your local music library.
• Listen to internet radio from Spotify, Grooveshark, Last.fm, SomaFM,Magnatune, Jamendo, SKY.fm, Digitally Imported, JAZZRADIO.com,Soundcloud, and Icecast.
• Search and play songs you've uploaded to Google Drive.
• Create smart playlists and dynamic playlists.
• Tabbed playlists, import and export M3U, XSPF, PLS and ASX.
• CUE sheet support.
• Play audio CDs."

(Link)

Pinta | open source paint program

Pinta is a free, open source drawing/editing program modeled after Paint.NET. Its goal is to provide users with a simple yet powerful way to draw and manipulate images on Linux, Mac, and Windows.

(Link)

Calibre | ebook management software

calibre is a free and open source e-book library management application developed by users of e-books for users of e-books. It has a cornucopia of features divided into the following main categories:

• Library Management E-book conversion 
• Syncing to e-book reader devices 
• Downloading news from the web and converting it into e-book form 
• Comprehensive e-book viewer 
• Content server for online access to your book collection

(Link)

UFRaw | raw image editor

"UFRaw (which stands for Unidentified Flying Raw) is an application which can read and manipulate photographs in raw image formats, as created by many digital cameras. UFRaw used to be available both as a stand-alone program and as a GIMP plugin. However, since UFRaw-0.19, the GIMP plugin is no longer officially supported. As a stand-alone program, UFRaw can be invoked with a graphical interface, or as a command line batch processing utility.[1] UFRaw reads raw images, using dcraw as a back end, and supports color management via LittleCMS, allowing the user to apply input, output, and display color profiles (see also Linux color management). Thanks to dcraw's versatility, UFRaw supports nearly all of the numerous raw image formats used by digital camera manufacturers." [1]
(Link)

Dia | Diagram editor

Dia is roughly inspired by the commercial Windows program 'Visio,' though more geared towards informal diagrams for casual use. It can be used to draw many different kinds of diagrams. It currently has special objects to help draw entity relationship diagrams, UML diagrams, flowcharts, network diagrams, and many other diagrams. It is also possible to add support for new shapes by writing simple XML files, using a subset of SVG to draw the shape.

It can load and save diagrams to a custom XML format (gzipped by default, to save space), can export diagrams to a number of formats, including EPS, SVG, XFIG, WMF and PNG, and can print diagrams (including ones that span multiple pages).

Xmind | mind-mapping software

"XMind is a brainstorming and mind mapping software tool developed by XMind Ltd. of Hong Kong. The program is intended to assist users in capturing ideas, organizing various charts, and share them with collaboration. It supports mind maps, Ishikawa diagrams (also called fishbone diagrams or cause-and-effect diagrams), tree diagrams, organization charts, and spreadsheets. It can be used for knowledge management, meeting minutes, task management, and GTD." [1]

Besides the Mind Map structure, XMind also offers Org-chart, Tree-chart, Logic-chart, and etc. These charts will play an important role in some circumstances. For example, Org-chart can show the company/department/team structure clearly. Logic-chart is very useful in sorting the reasons. More important, all these charts can be used within one map! Every branch can have the most proper structure, even every topic. [2]

Freemind | mind-mapping software

"FreeMind is a free mind mapping application written in Java. FreeMind is licensed under the GNU General Public License. It provides extensive export capabilities. It runs on Microsoft Windows, Linux and Mac OS X via the Java Runtime Environment.

As with other mind mapping software packages, FreeMind allows the user to edit a hierarchical set of ideas around a central concept. The non-linear approach assists in brainstorming new outlines and projects as ideas are added around the mind map. As a Java application, FreeMind is portable across multiple platforms and retains the same user interface, causing some amount of variation from the common interface on each platform." (Wikipedia)
(Link)

Mahara | ePortfolios

Mahara is an open source ePortfolio and social networking web application originally created by the government of New Zealand. It provides users with tools to create and maintain a digital portfolio of their learning, and social networking features to allow users to interact with each other. Mahara content management system provides users with blogs, a resume builder, a file manager and a view creator - a tool to help users create arrangements of their content in a particular way for others to see.

The eportfolio site http://myportfolio.school.nz, provided by the New Zealand Ministry of Education, free-of-charge for all New Zealand schools, is powered by Mahara

VLC | Media Player

This is the Swiss Army Knife of video players. It will play just about anything you can throw at it:

"Simple, fast and powerful media player:

• Plays everything: Files, Discs, Webcams, Devices and Streams. 
• Plays most codecs with no codec packs needed:
• MPEG-2, DivX, H.264, MKV, WebM, WMV, MP3... 
• Runs on all platforms: Windows, Linux, Mac OS X, Unix... 
• Completely Free, 0 spyware, 0 ads and no user tracking. 
• Can do media conversion and streaming."

(Link)

Koha | Open source library management system

"Koha is the first free software library automation package. In use worldwide, its development is steered by a growing community of users collaborating to achieve their technology goals. Koha’s feature set continues to evolve and expand to meet the needs of its user base.

In use worldwide in libraries of all sizes, Koha is a true enterprise-class ILS with comprehensive functionality including basic or advanced options. Koha includes modules for circulation, cataloging, acquisitions, serials, reserves, patron management, branch relationships, and more."

(Link)

Geogebra | open source Mathematics software

What is GeoGebra?
GeoGebra is free and multi-platform dynamic mathematics software for all levels of education that joins geometry, algebra, tables, graphing, statistics and calculus in one easy-to-use package. It has received several educational software awards in Europe and the USA.

Quick Facts

• Graphics, algebra and tables are connected and fully dynamic
• Easy-to-use interface, yet many powerful features
• Authoring tool to create interactive learning materials as web pages
• Available in many languages for our millions of users around the world
• Open source software freely available for non-commercial users
  (Link)

Scribus | Desktop Publishing

Easy-to-use, powerful desktop publishing that allows people to produce professional-quality brochures, magazine layouts, newspapers, advertisements etc. easily from just about any computer.

"Scribus is an Open Source program that brings professional page layout and desktop publishing to Linux, Mac OS X, and Windows with a combination of press-ready output and new approaches to page design.

Underneath a modern and user-friendly interface, Scribus supports professional publishing features, such as color separations, CMYK and spot colors, ICC color management, and versatile PDF creation." (Link)

Jokosha | open source Garageband alternative

Jokosher is a simple and powerful multi-track studio. Jokosher provides a complete application for recording, editing, mixing and exporting audio, and has been specifically designed with usability in mind. The developers behind Jokosher have re-thought audio production at every level, and created something devilishly simple to use. (Link)

Stellarium | Stars, Planets, Constellations

Stellarium allows you to place yourself anywhere in the world to watch the constellations travel across the sky above. Stellarium also allows you to pause, fast-forward or rewind time to see what the stars do. (Link)

There are some very cool videos on youtube that showcase what it can do:

Editshare Lightworks | High-end open source video editing

"Some of the finest films of the past 20 years have been edited with Lightworks. We’re proud of its heritage and we want to share it with you… for free. Discover why Oscar-winning editors across the world use it every day. Lightworks is the fastest, most accessible and focused NLE in the industry, because it is based on the simple idea that the editor, not the computer industry, knows what’s best. The latest release of Lightworks is based on the cumulative knowledge from twenty years of top-flight editing." (Link)

Celestia | Explore the universe

"The free space simulation that lets you explore our universe in three dimensions. Celestia runs on Windows, Linux, and Mac OS X.

• Unlike most planetarium software, Celestia doesn't confine you to the surface of the Earth. You can travel throughout the solar system, to any of over 100,000 stars, or even beyond the galaxy.
• All movement in Celestia is seamless; the exponential zoom feature lets you explore space across a huge range of scales, from galaxy clusters down to spacecraft only a few meters across. A 'point-and-goto' interface makes it simple to navigate through the universe to the object you want to visit.
• Celestia is expandable. Celestia comes with a large catalog of stars, galaxies, planets, moons, asteroids, comets, and spacecraft. If that's not enough, you can download dozens of easy to install add-ons with more objects." (Link)

Inkscape | Vector Graphic Editor

"An Open Source vector graphics editor, with capabilities similar to Illustrator, CorelDraw, or Xara X, using the W3C standard Scalable Vector Graphics (SVG) file format.

Inkscape supports many advanced SVG features (markers, clones, alpha blending, etc.) and great care is taken in designing a streamlined interface. It is very easy to edit nodes, perform complex path operations, trace bitmaps and much more. We also aim to maintain a thriving user and developer community by using open, community-oriented development." (Link)

Libre Office | Free Productivity / Office Suite

"LibreOffice is a comprehensive, professional-quality productivity suite that you can download and install for free, with no fear of copyright infringement. There is a large base of satisfied LibreOffice users worldwide, and it is available in more than 30 languages and for all major operating systems, including Microsoft Windows, Mac OS X and GNU/Linux.

What's outstanding about LibreOffice?

• LibreOffice is a feature-packed and mature desktop productivity package with some really great advantages:
• It's free – no worry about license costs or annual fees.
• No language barriers – it's available in a large number of languages, with more being added continually.
• Supports large number of file formats, including Microsoft Office, OOXML, and RTF – both loading and saving."

Flashcards | Anki

"Anki is a program which makes remembering things easy. Because it's a lot more efficient than traditional study methods, you can either greatly decrease your time spent studying, or greatly increase the amount you learn.

Anyone who needs to remember things in their daily life can benefit from Anki. Since it is content-agnostic and supports images, audio, videos and scientific markup (via LaTeX), the possibilities are endless.
For example:

• Learning a language
• Studying for medical and law exams
• Memorizing people's names and faces
• Brushing up on geography
• Mastering long poems
• Even practicing guitar chords!"