Achievements link to a "pre-requisite type" that defines it as completed.
This is a known list of types as of: 3rd October, 2018
0: complete specific Legacy thing
1: do n things
2: complete n other achievements
3: achieve n levels as class
4: affix n materia to the same piece of gear
5: complete all n of these requirement_2 - 9 things
6: complete a specific quest
7: complete all specific hunting log entries
8: discover every location within...
9: complete any of these requirement_2 - 9 quests (?)
10: level your companion chocobo to rank n
11: achieve PVP rank n with a specific Grand Company
12: participate in n matches in The Feast
13: triumph in n matches in The Feast
14: complete a specific Trial
15: achieve rank n with a specific Beast Tribe
16: there is no type 16
17: participate in n Frontline matches
18: guide a specific Grand Company to n Frontline victories
19: triumph in n Frontline matches
20: attune to all aether currents in a specific area
21: obtain n minions
22: there is no type 22
23: complete all Verminion challenges
24: obtain a variety of anima weapon
25: finish top 10 in a Regional Feast Championship