DRL's answer is a good summary of pre-amp/power-amp/rectifier tubes. Here are some specific examples of tubes used and how they affect tone.
For the pre-amp, far and away the most popular tubes used are 12AX7's, although sometimes you see 12AT7's used to drive the reverb or effects loop (if the amp has these features).
For the power amp section, the tubes used fall into two camps:
- 6V6 & 6L6: Fender is known for using 6V6's in their smaller amps and 6L6's in their larger amps. Either way, these tubes have a lot of clean headroom, so you get more volume from the amp before they start to distort. When they do break up, however, the sound can sometimes be a bit harsh.
- EL34 & EL84: Marshall is known for using EL34's and EL84's in their heads. These have less clean headroom, so they'll distort at lower volumes, but the distortion is usually warmer and smoother.
This explains why, historically speaking, Fender amps were/are popular with country/western guitarists who wanted a cleaner tone, while Marshall heads were/are often used by rock musicians wanting a great distorted tone (as with all generalizations, of course, there are exceptions).
Another factor involving tubes and tone include whether the power amp uses a Class A or Class A/B circuit design. A Class A design is typically less efficient and thus produces less volume, but with (subjectively, of course) better tone. Vox amps and smaller Fender tube amps often are Class A. Class A/B amps get more volume due to greater efficiency, but their sound isn't quite as sweet. Larger Fender amps and Marshall amps are usually Class A/B.
Finally, a hybrid amp will typically use tubes for the pre-amp section, but a solid-state power amp. The idea here is to use tubes for the portion of the amp that affects the tone the most, and then save money and weight in the power amp section, which is designed to be as clean as possible. Of course, these amps can't provide the really wonderful power-amp distortion of a great all-tube head set to 11, but that's the trade-off.