A level 77 player will be able to see 16 dig sites, 4 each continent if his Archaeology skill is high enough. The number of zones and continents you can find dig sites depends on your Archaeology skill and your character level. You have to dig one out to get a new dig site. There are always 4 dig sites per continent only 3 for the Broken Isleand it won't change on logout, server restart or anything. Once you train Archaeology, you will be able to see dig sites on your World Map. It's a good idea to level up Archaeology with at least one character. Just imagine your alt hitting level 91, and you can equip right away keep in mind, these are not Heirlooms, these will not scale with your character level.
However, most of the armors and weapons you can get are bind to account, which is really cool. You won't gain any real end-game benefits from Archaeology, most of the items you can get have no in-game power like pets, toys, and mounts. This Archaeology guide will try to teach you the basics of this profession.