What were the achievements in Vanilla World of Warcraft?

Achievements for vanilla WoW appear under the "legacy" tab in the achievements list. Most of them were retroactively awarded to players once the achievement system was implemented.

I have not been able to find a full list of legacy achievements specific to vanilla. However, you can find vanilla achievements easily enough by searching the wowhead database for legacy achievements with "60" or "classic" in their description. I cannot guarantee that all of them have this identifier though.

Below are links to all of the legacy achievement lists on WoWhead:

General

Dungeons

Raids

Professions

Player vs. Player

Currency

Expansion Features

Scroll through these or use Ctrl+F and look for "60" or "classic". Off the top of my head, I know you'll find some vanilla achievements related to pvp, dungeons, and raids. For example, if you browse the legacy PvP achievements and search for "60", you will find achievements that rewarded titles based on the highest achieved PvP rank during vanilla, such as Blood Guard. There are also some dungeon and raid achievements, such as Onyxia.

Note: As mentioned in the comments, some of these achievements were actually attainable after Vanilla.