Hang out in amp or guitar chat rooms or check out lots of manufacturers' promotional write-ups and you'll notice that that term "Class A" gets slung around more than hash off the griddle at a greasy spoon in Memphis. Call a thing "Class A" and it automatically sounds superior, right? As it applies to tube amps, though, Class A isn't necessarily on a par with Grade-A beef or First-Class mail. The designation might be significant if you're looking for a certain sound from an amp, but it doesn't make that amp better in and of itself.
Class A and Class AB are technical terms used to define the way a tube amp's output stage works, not qualitative assessments of their build quality or performance. Note, too, that this Class can only be defined through strict measurements on the work bench, and under such conditions many amps billed as "Class A" actually are not so — they are merely cathode-biased amps with no negative feedback, characteristics that have fallen under the "Class A" moniker for many players and manufacturers. Never mind, it's a slang term of sorts, but it still helps to designate the type of amp that carries the label.
So-called Class A amps include the legendary Vox AC15 and AC30, Marshall 18 watter, Fender's tweed Deluxe and some others, and several contemporary models, including many high-end amps inspired by the Vox and tweed Fender templates. Several of these are great amps, beloved for their chiming clean tones and high harmonic overtone content when pushed into distortion. If those are sounds you're looking for, the Class A label might be worth seeking out.
Many, many great amps, on the other hand, also carry the Class AB designation. This term applies to amps with two or more output tubes in a push-pull configuration, with either set or adjustable fixed biasing, and usually a negative feedback loop at the output stage. (Sorry for all the technical jargon.) Great Class AB amps include Fender's tweed Bassman, blackface Deluxe Reverb and Twin Reverb, Marshall's plexi and metal-panel heads, all the big HiWatts, the Mesa/Boogie Mark I and others, Soldano's SLO, Bogner's Shiva, and so many excellent amps they're difficult to tally. Characteristics of the Class AB amp can include a firmer, punchier tone, excellent articulation in clean sounds, and an aggressive distortion when pushed hard. That said, both types of amps will crossover to many of the typical traits of the other species, so none of these descriptions are hard and fast. If you're searching for a new amp, or just wanting to learn more about what makes them tick, these Class designations might be of interest, but they should never be taken as the be-all and end-all factor of your amp-buying decision.