norm tends to use titebond, as alluded to above; there are various flavours of titebond, all with different properties, but at the end of the day, titebond is just a brand name, and the glue TYPE is what metters.
General PVA will do for most applications, with some caveats. It's not waterproof, even the so called outdoor flavours; its grab time is not particularly quick, which can be both a good and a bad thing. its set time is long, typically 24 hours for full cure. But, once cured, a decent quality pva will be stronger than the wood itself.
Polyurethane (like gorilla, or titebond III (I think)) glues - waterproof, fast grab and cure time. Gap filling, although the gap fill has no strength, so shouldn't be used to fix a sloppy joint. Stains fingers badly! Limited shelf life. Cures by absorbing water from the surrounds, so not good for biscuit joinery, cos biscuits work by also absorbing water to swell and fill the slots.
Traditional hide glues - don't know much about these, but 1 main advantage is that joints glued with this can be split at a later date, for repairs or whatever, by exposing to steam. Which, of course, indicates they're not waterproof, and not suited to humid environment like kitchen or bathroom.
There are all the contact glues out there for 'instant' grab and cure - limited use in woodworking, I'd have thought.
Extramite is a powdered glue (mix with water) with a reasonable open time and pretty quick cure time - as in 'kick around workshop within half hour' - useful if time's short, or you're fed up of glueing up and having to wait 24 hours to move on to next stage. I think you may be able to get this pre-mixed too.
Urea-formaldehyde (UF) is another major type of glue, about whiich I know nothing.
Anyway - if you want a general purpose, indoor use glue, PVA is the way to go. And there ARE flavours out there with shortened grab and cure times for the impatient.
[EDIT - extramite (powder form) is a UF type of glue, i've just found]