Open source software is software that is free and the source code is open for anyone to modify, which means that many people may have input in how the software operates. There are generally no licensing restrictions – you can use and distribute it freely.
Freeware is software that can be used freely. The author owns the software and users cannot modify it. There may be restrictions about sharing it.
Shareware is trial software that can be evaluated for a specific time (generally 30 days) for free. Often, the software stops functioning at the end of the evaluation period and the user must purchase to re-activate it.

Open Source Software
Open Office - MS Office alternative
GIMP- Photoshop alternative (download both GIMP and GIMP Runtime Environment)
Photo Gadget - for resizing photos
Tux Paint - kid-friendly paint program
Tux Math - math game
CutePDF - convert files to PDF
Typefaster Typing Tutor
Sqirlz Morph
Audacity - edit sound files - install the software and the LAME file if you want to save music as MP3s
Cleanup! - keep your computer clean
CCleaner - keep your computer clean

Wikipedia's Long List of Open Source Software
SchoolForge's List