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Software

    There is an astounding amount of free software (often open-source) available on the internet.
    What follows is a list of free software I find particularly useful:

   Mathematics:

Wolfram|Alpha

Computer Algebra System (online)
http://www.wolframalpha.com/
A "computational knowledge engine". However, does a lot more than just math; see http://www.wolframalpha.com/examples/ for more examples. Does not require most questions to be formatted according to any particular syntax. For computations requiring multiple steps or more flexibility, try Maxima.
Maxima
Computer Algebra System (download)
http://maxima.sourceforge.net/
Handles numeric and symbolic mathematical computations. Unlike W|A, requires some knowledge of syntax; many tutorials available. Work can easily be saved and loaded at a later time.  The program wxMaxima is included and recommended for a nicely-formatted interface.
GeoGebra
Geometry/Algebra/Spreadsheet (online & offline)
http://www.geogebra.org/cms/
Create interactive Euclidean geometry constructions or analytic (coordinate) geometry constructions; also creates interactive diagrams with Calculus content.  Includes spreadsheet.  Can be used through website or downloaded.
R
Statistics Software (offline)
MiKTeX Mathematical Typesetting (back-end) (offline)
TeXMaker Mathematical Typesetting (front-end) (offline)
ScribTeX
Mathematical Typesetting (online)
MathJax Mathematical Typesetting (online)
asciiSVG Mathematical Graphics (online & offline)
LiveGraphics3D Mathematical Graphics (online and offline)
http://www.vis.uni-stuttgart.de/~kraus/LiveGraphics3D/

For an introduction to using this program, see this article; for an example of this program in action, see this webpage.

   Computer Science:

Java Programming Language (online and download)
http://www.java.com/

BlueJ Integrated Development Environment for Java (download)
http://bluej.org/
Notepad++ Text Editor with Syntax Highlighting (download)
http://notepad-plus-plus.org/
IDEone Code Compiler and Storage (online)
http://www.ideone.com
An online compiler and debugging tool which allows you to compile and run code online in more than 40 programming languages; highlights syntax; code is automatically saved and a link is generated.
For example (Java):http://ideone.com/cE4RR 
If you register an account, you can save your code.
NetLogo Agent-Based Modeling Software (download)
http://ccl.northwestern.edu/netlogo/
AScape Agent-Based Modeling Toolkit (download)
http://ascape.sourceforge.net/

Repast Agent-Based Modeling Toolkit (download)
http://repast.sourceforge.net/repast_3/index.html

jMonkeyEngine Three-Dimensional Programming Toolkit (download)
http://www.jmonkeyengine.com/

   Additional Applications:

Google
OpenOffice
GIMP
Art of Illusion
DropBox
PortableApps
USB Linux
VirtuaWin
Jing
SWF to AVI
Screenr
Prezi
FS Capture Screen Capture Tool (download)
http://www.aplusfreeware.com/categories/mmedia/FastStoneCapture.html
Mindomo Mindmap Creator (online)
Gliffy Flowchart Creator (online)
Jarnal
Notetaker and PDF Annotator (download)
http://jarnal.wikispaces.com/
Write, sketch, or annotate a document using a stylus, mouse, or keyboard.  (Particularly useful for stylus input.)  Includes handwriting recognition.
ImageJ Image Analysis (download)
http://rsbweb.nih.gov/ij/
Display, edit, and analyze image files.  Calculate lengths, angles, and areas of selected regions; detect edges; count particles; perform geometric transformations.
Audacity Sound Recorder and Editor (download)
http://audacity.sourceforge.net/
Free, open source software for recording and editing sound files.  Create multi-track recordings, import and export various formats, add effects, change sample rates.
LibraryThing Personal Book Catalog (online)
http://www.librarything.com
Organizes information about your personal book collection; provides a social network for book lovers.  For example, to see what's on the shelves in my office:
http://www.librarything.com/catalog.php?view=stemkoski&shelf=shelf&sort=tags
OnlineClock An Online Clock (online)
http://onlineclock.net/
A simple digital clock on a plain background; clock can be enlarged; convenient to have in the front of the classroom when students are taking exams.