Free Typefaces

There are a number of quality free typefaces available. By no means is this an exhaustive list of what is available. These are some of the better places to find them, though.
Greek Font Society: The Greek Font Society exists to provide and promote typefaces for the Greek-speaking audience. Having said that, many of their typefaces come with a full Roman complement as well. GFS Elpis (by Natasha Raissaki) is a personal favorite.
Gentium: Gentium, a project by Victor Gaultney undertaken by SIL, is a highly useful and quite attractive serif face. Until GFS Elpis came to my awareness, it was my main font. At Products by SIL there are a number of multilingual fonts and resources.
Exljbris: Exljbris advertises itself as a "free quality font foundry", and, from what I have seen, Jos Buivenga's fonts certainly live up to advance billing. Calluna has now replaced GFS Elpis as my go-to font.
Manfred Klein's Fonts: Klein's pictographic fonts inspired Kelly's first release, LutheranPics.
Iconian: Dan Zadorozny has spent a lot of time creating lighthearted fonts, and would like folks like you and me to use them.
Typophile Free Fonts FAQ: This answers a lot of questions about free fonts and gives some reputable sources for them.

Paid Typefaces

There are many good sites to buy fonts from on the web. Monotype and Bitstream seems to have ownership of several.
MyFonts: This is the widest ranging paid-font site. Pretty much anyone can get their fonts on here, but only the good make money. A corporate site, but with a good selection.

Type Designers

Unlike me, these are type designers who know what they are doing, and do it well.
Shinn Type: Toronto-based type designer Nick Shinn. Among his creations are the fonts used in the Globe And Mail.
Nick's Fonts: The fonts of Nick Curtis--more "fun" oriented. Great quality.
Tiro Typeworks: John Hudson and Ross Mills live in one of the most beautiful parts of Canada, "my home and native land", and have the fonts to prove it.

Type-Related Resources

One could conceivably get away with placing but one link in this category, but I'll put more than that, for the sake of anyone other than me who uses this site.
On Snot and Fonts: Luc Devroye has apparently spent more time Googleing and Yahoo!ing fonts than anyone else on the planet. Always opinionated, sometimes even offensive, but about as comprehensive a set of links to fonts and font-related resources as you will find anywhere. My font searches always begin with Luc's site.
Typophile: The place to find out what's happening in the world of fonts. Many top professional typeface designers hang out here.
I Love Typography: For all of you who love the way what you read looks.

Font Design Software

I have played with all the Windows-based software listed below. Currently, we use High-Logic's Font Creator Program because of one particular feature--the ability to directly import from line-drawings. TypeTool has a better drawing interface, and FontForge is way more powerful, though also really hard to set up. I'd suggest you try each of them out before buying anything. FontForge may be all you need and want, and it's free.
Font Creator Program: This program is able to import from BMP files. It does a passable job of directly rendering what you drew into vectors. Some post-production work is necessary, but it is a lot easier than hand-tracing, especially on very detailed sketches, such as in Kelly's first picture font.
TypeTool: This introductory font creating package is based on the venerable FontLab package, lacking only some of the more advanced features. An excellent program, especially at its price. At the moment, we could only afford one, so we went for FCP. But I wouldn't be surprised if in a year or two we added TypeTool to our design tools.
FontForge: This is a very powerful font design package, and, a SourceForge project as well. Not recommended for anyone who is unfamiliar with CygWin and running XWindows emulated on top of Windows. It does have the advantage of being free, as in BSD-licensed free.
Type 3.2: This is a fairly straightforward and simple type design package. It can edit either PS or TT outlines. Fairly inexpensive. The free version is also very interesting.
FontStruct: Kind of like a font-building Tinkertoy set, this site allows you to create TTF fonts using a grid and some standard building materials. You won't design a masterpiece, but you might just have the most fun yet making a font from scratch. Requires Flash.

Last updated March 2017 by the webmaster. These links originally compiled typing one-handed with a thrashing infant on the webmaster's lap, and updated with that same infant sitting nearby. And now finally updated nearly 9 years later with said infant no longer an infant but a child nearing teenagehood... sigh...