I bought a Trek FX 7.4 last summer, and now, as the conditions are getting more and more slippery, I'm thinking of getting a bit more winter-y tires. The rims are tubeless, but I'm a bit confused of what kind of tires I can buy. Are all tires compatible with these tubeless rims? Do I need to fit an inner tube with some tires, or do all tires work "tubeless"?
The rims spec'd with a lot of new Trek bikes (including yours) are "Tubeless ready". This means that you can run tubeless, but you can also run regular old tires with tubes (i.e. clinchers).
The primary advantage of running a tubeless tire (which will be marked as tubeless) is that you can run a lower pressure without getting pinch flats, which is advantageous in the winter. However, you have to faff about with sealant (and the right valves, rim strips) to get this done [along with obviously buying the tubeless tire]. Note that if the seal fails, you should be carrying an inner tube (regardless of running a regular or tubeless tire) so you can throw it into the tire to get you home.
If you run a regular tire (i.e. a clincher -- tubular tires are a specialty item for track racers, so don't get the terms confused), you should an inner tube in and install as normal. If you're running the tires that shipped with the 7.4 FX originally, you have a regular tire (which has a tube), so you just find another tire which is about the same size (probably something around 700x32), pop off the old one and pop on a new one. Regular tires are also a bit cheaper.
There are people who run regular tires as tubeless (at their own risk - some tires take this, others don't). You can try this at your own risk, but for winter, I'd probably just go for a regular clincher tire with inner tube. The tires you can fit onto something like a 7.4 FX will be narrow enough that the lower pressure advantage of a tubeless tire will not really help you. Maybe a regular studded tire depending on where you are and what your road conditions are like.
Many tyres can be made to work without a tube, but tubeless tyres tend to work fine with tubes.
The bead of a tubeless tyre is smooth and moulded more accurately than non-tubeless tyres, to form a tighter seal with tubeless-compatible rims. You may be able to achieve tubeless running with a non-tubeless tyre, but it's likely you will need to use more sealant compound.
Some tubeless systems use a standalone valve, others use a valve integrated into the rim strip. For "ghetto tubeless", you can take an innertube from a smaller size and cut it to resemble the tubeless rim strips that have valves. (I've never done this, YMMV).
If you don't mind getting messy with sealant, try it. If you fail, fit a tube!