There are some terms that are used by the Mozilla community. One sees them on IRC or in bugs. It can be helpful to know exactly what these refer to. Note that this is an attempt at describing the current usage of these terms. This usage may change.
Cutting edge Firefox downloads. As a development organization, the Mozilla community tends to describe things from the bottom up. So to figure out how to download a cutting edge or bleeding edge or 'beta' version of Firefox, you need to look for a "build" (which is developer-speak for the packaged files you can download) of 1.9.1 (the number of the underlying 'platform' called 'Gecko' or 'mozilla' that Firefox uses). As it turns out there is a platform number for every Firefox number, but they are not the same number. Firefox 3.0 corresponds to mozilla 1.9.0; Firefox 3.5 corresponds to mozilla 1.9.1; Firefox 3.6 corresponds to mozilla 1.9.2. However, the numbers are not used all of the time. Sometimes the Firefox releases are described by a code name. For example "Shiretoko" is used for Firefox 3.5. And very often neither names nor numbers are used, but rather special words indicating relative order are used. The three common cases are "trunk", meaning the most recent version of mozilla; "branch", meaning the version before trunk (though it can be any version before trunk); and "next", meaning a possible version in the future. So, if you are looking for the cutting edge Firefox, you probably want a mozilla branch nightly build, that is, look for a mozilla thing (not a Firefox thing), not the breaky one on trunk, but the most recent branch, and built very fresh (last night!), so you can install it. For Firefox 3.5 that would be latest-mozilla-1.9.1 in https://archive.mozilla.org/pub/firefox/nightly/.
trunk build. A trunk build is a build made off the trunk, i.e., the latest, newest, but also least stable and most buggy state of the source. Trunk builds are provided so that one can execute tests without going to the trouble of building the source. The current trunk is mozilla 1.9.2.
tinderbox builds. For the trunk and the latest two branches, there are machines at Mozilla and maybe elsewhere compiling builds as fast as they can make them for at least all three of Firefox, Thunderbird and SeaMonkey on all three of Linux-i686, Mac-UniversalBinary and Win32. These are the "tinderbox builds", also known as "hourly builds" though it usually takes more than one hour to make one; they are followed by automatic tests and their main purpose is to check that nothing is horridly wrong with the latest change to the source: what they are doing is to constantly check that the current source can be built into an executable and that that executable passes a certain more-or-less fixed set of tests.
nightly builds. The same machines produce about every 24 hours a "nightly" build which can be downloaded by anyone if you know where to get it: these are the nightly builds, and testers all over the world download them and test them, reporting as they go along on any bugs that hit them. Anyone can be a tester, you don't need to get approval before you can start testing. When people talk of "nightlies", they often mean "trunk nightlies", though there exist also "branch nightlies" which are somewhat less buggy but also not as "newest & greatest". For instance, one may use a Thunderbird "branch nightly" as one's main e-mail client.
Note that nightly builds may not be created for every platform for every night. If one has questions about the way a particular nightly was built, the best way to get that information is to download the executable, launch it, and then go to the "about:buildConfig" page, by typing this into the Location Bar. This should provide all the build flags and configuration settings used for building this executable.
Nightly builds can be downloaded from https://archive.mozilla.org/pub/firefox/nightly/.
If you need to download builds often, most builds can be downloaded semi-automatically and saved with a convenient name using mozdownload. See also How to do regression testing.