In my opinion, there are two basic types of archtop guitars. The first type is designed to be an acoustic instrument - designed, voiced and constructed to have a certain acoustic sound.
The acoustic archtop was originally developed by Gibson with significant early contributions from Lloyd Loar. They designed an acoustic guitar that was louder and more focused (fewer acoustic overtones) than the traditional flattop. They were used in orchestras and big bands (think Freddie Green, Eddie Lang, etc.) and primarily played a 4 to the bar cadence.
Interestingly, the chord voicings utilized were 3 or 4 note chord shapes instead of the full, 6 string barre chords. Freddie Green rarely played more than 3 note chords and in some cases only 2 notes. His 4 to the bar strum locked in with the bass player.
The design of the early archtops produced a clear, bright, cutting sound, less resonant than the sound produced by a flattop. Carved spruce or maple tops, x bracing, large bodies (17" or 18" at the lower bout) high action (string height), offset neck (to further raise the strings off the top), trapeze tailpiece and voilin style floating bridge all contributed to the volume and clarity of the original archtops.
In my opinion, the second type of archtop is the electric archtop. These are archtops that are designed primarily as an electric instrument AND whose acoustic characteristics are judged primarily by how they will translate into an electric sound. For example, perhaps the quintessential modern day electric archtop is the Gibson ES175. Laminated top and back plates combined with parallel bracing produce a very focused, bright acoustic tone. Not pretty in and of its unplugged self but it produces acoustic characteristics that translate nicely to an amplified sound. Combined with a top mounted PAF humbucker or a single coil P90 and you get the classic electric jazz guitar sound. Gibson achieved a design that could be played at higher volumes (i.e. through an amp) and that would resist feedback while producing a warm, controlled amplified sound.
Voila...practically every known jazz guitarist on the planet has played an ES 175 at some point in their career - Kenny Burrell, Jimmy Raney, Joe Pass, Herb Ellis, Jim Hall and Pat Metheny to name just a few.
Although you can divide archtops into 2 basic groups (acoustic and electric), these days you see varieties that fit the traditional definitions as well as all flavors in between.
By the way, have I ever mentioned that I secretly hope Taylor will some day add an archtop guitar to their line???
My 2 cents.