If you're talking about "brew day" time, that's influenced much more by your brewing equipment and process (i.e. do you do all-grain or extract? fly sparge / batch sparge / brew-in-a-bag?). This typically doesn't change much from one type of beer to another (with a few exceptional cases where a particular style may require an extra step or two that will make it take longer than usual)
If you're talking about fermentation time, that can depend on a lot of variables, many of which are specific to the beer style:
- Temperature: Generally, the warmer the fermentation, the faster it goes (within reason). For this reason, ales usually go faster than lagers, since they're intended to ferment at a higher temperature.
- Gravity of the wort: The higher your OG, the longer the fermentation will take. This is both because the environment is more stressful for the yeast, and just because there's a lot more sugar for them to work through before they're done. Obviously, this means lighter beers will typically ferment faster than big, heavy, highly alcoholic ones.
- Strain of yeast: This is an important one that's often overlooked. Not all yeasts are created equal, and some will just naturally be more vigorous than others. This also ties in a bit with temperature, as some yeast strains work better at higher temperatures (and produce fewer off-flavors, etc) so you can "run them a bit hotter" if you need to.
- Oxygen and other nutrients: The happier your yeast are, the more they'll reproduce, and the faster the whole process goes. This is usually more a general "equipment and procedures" issue than a "style of beer" thing, but there are some styles of beer (with a lot of non-malt additions) which may be somewhat deficient in some nutrients, which can cause them to run slower if not otherwise supplemented.
- Post-fermentation conditioning: Many styles of beer require lagering or cold-contitioning after fermentation to get an acceptable level of clarity or to drive off certain off flavors. Others just take a while in the keg or bottle before all their flavors really "meld" together to really taste their best. In general, lagers will need some extra time for this, and heavier or more alcoholic beers often require more mellowing time after fermentation.
It's kinda hard to pin down 2 or 3 specific styles, because often there's a lot of variation depending on the brewer, the location, and other factors which might make different people's results vary, but what I can tell you is based on the above, if you're looking for quicker styles of beer to get a finished product, you're generally going to be looking for:
- An ale
- On the lighter side (i.e. a "session" beer, etc)
- Using either a highly-flocculating yeast, or a style where cloudiness isn't a big deal (so you don't have to condition it for clarity)
Given this, wheat beers are probably a good choice (typically lighter ales and they're supposed to be cloudy). I've also had some nice quick brews when working with some British/Irish style ales, as the esters from higher temperatures are often part of the style, and the yeast strains are often fairly vigorous (I just did an Irish Red that finished fermenting in about 5 days).
BYO actually has an article specifically on this topic as well, with some recipes, which you may want to check out: https://byo.com/aging/item/1397-speed-brewing