Free Software

By the time you pay for computer hardware, you've already sunk a lot of money into your computing "experience". You might as well use free software whenever possible to keep down your total costs. In some cases, the free alternatives are even superior to their non-free counterparts. Here are some free programs I use and some notes about them. I use open source software whenever possible, and freeware if necessary. I never use shareware that needs to be registered.

Audio
Audacity Very well done waveform editor. Allows for recording, mixing of tracks, adding effects, and is extensible through various plugins. I use it to trim up audio files and normalize volumes, but it's more powerful than that. My only complaint is that the menu layout could be improved. You'll need an MP3 encoder to be able export MP3s. Open source under the GPL.
CDex There are a ton of programs that will rip CDs, many as part of other programs. CDex is the best standalone one. You can rip CD audio to either WAV or MP3, re-encode existing audio files, and go from WAV to MP3 or vice-versa. CDDB support makes tagging easy. It works great out of the box, but upgrade the MP3 encoder to a later version for even better quality. Open source under the GPL.
MP3 Tag Tools As the name implies, this is for editing a bunch of MP3 ID3 tags at once. It does that quite admirably. I don't like how you have to enter something for the "track" part of the tag, but otherwise it's great. GPL.
Winamp Freeware, not open source, but it gets my recommendation. If you haven't seen it, it's a media player with a good equalizer, visualization system, and playlist editor. Through plugins it can play pretty much anything. I think the video player and Media Library are unnecessary features, but otherwise this program is top quality. Easily the best media player out there.
Songbird Songbird is starting to challenge Winamp as my preferred music player, but it's not quite there yet. It's killer feature is that it's basically Firefox with a media player attached. You can listen to linked or embeddedmusic files from within the browser, which is way cool. It's free and open source.
Graphics
The GIMP The Windows port of the best Linux graphics manipulator. Think of a cheap answer to Adobe's PhotoShop. Despite some fanatics' claims, it can't replace PhotoShop for a skilled user, but for most purposes it gets the job done. There are plenty of effects, filters, and tools to play with. Gripes? It isn't stable on Windows. Save early and often. The menus aren't grouped all that well. Since the 2.4 release, it's improved by leaps and bounds. If you don't have a real photo editor and don't want to cough up the money for one, you can't beat GIMP. Open source under the GPL.
Paint.NET Paint.NET is a handy Windows-only tool that I use sometimes when the GIMP seems to be overkill. It's very pretty and very stable, and it's a good way to quickly touch up pictures. Open source under an MIT license.
Instant Messengers
Pidgin There used to be some debate as to what the best multiple-interface Instant Messenger client was. Now there is not. Pidgin can connect to AIM, MSN Messenger, Google Talk, and scads of others. I use it all the time and highly recommend it. Open source under the GPL.
Internet
Azureus High speed internet users only. Azureus is a BitTorrent protocol client written in Java, so it's cross platform. If you've never heard of it, BitTorrent is a file transfer protocol that can achieve very high download speeds by grabbing pieces of a desired file from many different users. You're required to allow others to get pieces from you in order to get good download speeds. I use it for downloading Linux distributions, but there are whole subcommunities. Open source under the GPL.
FileZilla There are several alternative FTP clients, but I like FileZilla best because it's open source, not shareware or freeware. It has a good, intuitive interface, and plenty of configuration options so it's possible to get it working from behind a proxy and firewall, unlike some other clients. It also doesn't touch the Windows registry, always a plus. The site manager stores logins for various sites, and it makes maintaining this web page a little easier. Open source.
Firefox The best web browser, bar none. Designed with simplicity and usability in mind, Firefox makes browsing the internet an enjoyable task again. Replace Internet Explorer with it as soon as you can. It blocks popups, features tabbed browsing, editable history, and better standards support than IE. It's features are nearly limitless if you factor in the extension system; scores of enhancements are already available. You can make it look nice through the theme system. Distributed under the Mozilla Public License, so it's free and the source is available.
FreeProxy A must if you have a home network. I prefer this to the standard Internet Connection sharing. Setup is a little involved, but once you're running you have can a fully functional HTTP, FTP, and SOCKS proxy over which to share an internet connection. You can also set up an e-mail proxy and HTTP server. Not open source, but free to use. Hopefully future versions will allow HTTP/ FTP over SOCKS. Not open source, but still nice and free.
Mozilla If you don't get Firefox, get the Mozilla Suite. It's a bigger download and clunkier interface, but it has the same popup blocking and tabbed browsing as Firefox. There's also an e-mail client, IRC client, HTML editor, and web developer tools. Mozilla Public License.
Kompozer A very good visual HTML editor that produces clean markup based on the Mozilla Suite Composer. Highly recommended. It still needs some user interface and CSS integration work before I start using it, but I recommend it to those who don't feel the need to muck with HTML when it's not necessary. This is a bugfix version of the official Nvu, which is very oudated, and works a little better. It's open source.
Putty A command-line tool that lets you access a remote system via SSH. I had to use it for school, and it beats the usual "telnet" app handily. GPL.
Thunderbird The e-mail counterpart to Firefox. It blocks spam. It doesn't have the security holes Outlook Express does. Get it. Spam blocking works quite well, but needs to be "trained" for the type of spam you receive. It's a little involved getting that set up, but after a week or so of marking junk manually it'll do pretty well blocking for you. If it worked better with my school's e-mail server, I would use it exclusively.
WinSCP WinSCP is a program that allows you to explore a remote file system in an Explorer-like interface. If you are looking for something to, say, access your student account at school, this is exactly what you need. 
File/ System
7-Zip Probably the best zip/ unzip/ compression/ decompression software. The 7zip format compresses even further than zip in most cases. You have to configure it to associate it with archives and give it a place int he context menu, but once you do, it's good to go. GPL.
Bochs A real nerd's tool, Bochs is an emulator of IA-32 processors... think of it like a computer running in a program on your computer. I use it to test bootable CDs without having to constantly reboot, and you can use it to try out low-footprint operating systems, like FreeDOS. It'd be nice if there were a GUI to configure settings instead of the text file, but this is a techy-only tool anyway. GPL.
ClamWin Clamwin open source anti-virus program for Windows. This sort of thing has been needed for a long time, and is finally becoming a reality. If you have any programming talents, see if you can help them out. ClamWin can scan your PC for viruses, but doesn't do "real-time" protection yet. GPL.
Spybot Download this right now. Spybot will help you get rid of advertising software that gets installed by malicious web sites and programs... if you're having problems with a slow computer, popup ads coming up all the time, connections being dialed, and weird system erros, you need this. Developed by a German college student and maintained by a community, this is possibly the most valuable tool on my CD I take around to fix computers. Download it. Update it. Use it regularly.
TightVNC TightVNC is a remote computing tool... it lets one see and control a computer on the network/ internet from the first computer, even if it's not of the same operating system. This is very valuable for long distance troubleshooting. GPL.
Uniflash Venturing into serious tech-head software, we have UniFlash. UniFlash is a program that can flash BIOS chips of just about anything... hopefully you don't need to do this often, but I've needed it several times before. It's especially handy for backing up flash ROMs and performing the dreaded hot-swap of BIOS chips.
Video
DScaler This is a video capture program for Bt829 chipset TV cards, of which I have three. I use it to watch TV, play videogames, and captures movies and screenshots. Getting it set up is a little bit of a trip, but it's gold once it gets working. It's easily better than most default TV tuner software.
VirtualDub If you need to do some basic video editing, this is the tool you need.
Other
Dev-C++ Utterly brilliant integrated development environment designed for C and C++ programming. It serves as text editor that will indent and color your source code to make it sort of make sense and help you catch errors, includes a compiler, a debugger, version-ing tools, and all kinds of stuff. Highly recommended.
Graph This is a neat GPL program that will graph math functions like a graphing calculator. Useful, since you can print them after that, unlike with my TI-89.
GraphCalc Like Graph, but this one does 3D graphs.
OpenOffice.org Microsoft Office is expensive. OpenOffice.org is free, and does the same things. It's a little slower and a little less mature, but is constantly being upgraded and improved. It can export to just about any document format, including PDF, which is incredibly handy. Its label creator is better than Word's, for what that's worth. In addition to word processing, it has a spreadsheet (like Excel), presentation (like PowerPoint), HTML editor, and limited drawing program.
PDF Creator There are a number of programs that will generate a file in Adobe's PDF format, but lots are A) Expensive, such as Adobe's versions, B) Annoying, such as the "printer driver" programs that leave a watermark in the resultant file or bug you with ads, or C) Not very useful. PDF Creator is none of these things. It's free, has no ads (it's GPL, in fact), and does lots of useful things. Not only can you generate PDFs from anything you can print, you can also choose to generate files in different image formats.
PDF Toolkit Builder Dealing with Adobe's PDF format becomes much more pleasant when you don't have to pay a bunch of money to edit the files. PDFTK Builder allows you to merge and split PDFs very simply. It's easy, free (GPL), and highly useful.