So far there are two types.
The counter type doesn't really do anything other than count the number of times you've died, been resurrected, used the /lol emote, flasks you drank, kills in pvp, highest strength, highest armor, etc. Nothing other than just mildly interesting for historical purposes or for showing off.
The second type however, awards you "achievement points". Completing each task will give you a score varying between 5-100 points. The exact purpose of the points (other than tracking the total) isn't clear yet. Perhaps you'll be able to spend the points at a later time to buy some items or a tabard, time will tell.
Achievements + Counter
Blizzard has stated that anything currently in game will be tracked retroactively. Meaning, total map explored, exalted factions, lifetime honorable kills, total vanity pets, bank slots, legendary items obtained, etc. Stats such as the total times /lol used, number of deaths by boss X, total number of raids, total number of guilds, are not possible to be applied in the same way. You'll be able to start earning these as soon as the 3.0 expansion patch is released.
I wouldn't care too much about the counters myself, the achievements should be fun to collect however. I know there'll be at least one person in our guild that'll for sure go out of his way to destroy 1.000.000 Tuskarr, just to get that achievement. He shall remain unnamed for now. But, I doubt there's a single person in our guild who doesn't know the answer to that... Which does give a new idea for a poll.
See deatknight.info, WoR or the the official blizzard page (posted 17th july 2008) for related screenshots and info on achievements.