There are a lot of different kinds. Some are really for a particular use and some are a good multi-purpose adhesive. Here is a partial list to guide you. :)
TYPES:
-glue sticks (widely available by many brands) - doesn't work very well on vellum and I don't like it very much on heavy stuff
-photo squares or mounting squares (double sided adhesive tape cut into squares, widely available by many brands) Here are some brands: Fiskars, Scrapbook Adhesives - this is a good multi-purpose item, would definitely have on hand
-tape runner (a hand held dispenser applies a "tape" of adhesive to the paper-permanent, removable, vellum..., widely available by many brands) Here are some names: Mono Adhesive Permanent Bond or Removable Bond (made by Tombo), 3L E - Z Runner, Herma, Therm O, EK Success - good all purpose adhesive, more costly but is quick to use
-glue dots (a dot of adhesive that is pretty strong, use for attaching heavy things, widely available by many brands) Here are some brands: Glue Dots, EK Success, Therm O, K & Co. - would use these to hold heavy items, like embellishments, I wouldn't use these for just paper - good to have on hand though!
-double sided tape (a roll of double sided tape, you can get permanent or re-positionable: when it isn't permanent-this can allow you to try an image in a certain place before committing to it's location) Here are some brands: Therm O, K & Co., Art Accentz - multi purpose adhesive but I like photo squares better
-glue pen (starts out blue and becomes clear) Made by Zig 2 - can use on vellum and a few other things, more of a specialty item
-vellum tape (for use on vellum paper) Here are some brands: Scotch, K & Co. - good to have on hand if you work with vellum a lot
-foam dots (adhesive foam shapes for raising images) Here are some brands: EK Success - great if you want to raise an image or item up off the paper
-photo mounting tape (just like regular tape but for scrapbooking) Brand: Scotch - another adhesive but you can just use photo squares
-and then, of course, there is a Xyron machine(s) that can turn anything relatively flat into a sticker, it can laminate, and it can magnetize items (it just depends on what machine you get)
I know that was a lot of information but I wanted to help you with the different kinds. There are still other types out there, but how many can a girl list?!?!? :) Hope this helps you. Happy Scrapping!