Once you read level 30 in one of the base classes, you have the ability to switch to a 'job': a more specialized version of the class you're playing.
All job requirements can be found here:
Bard = 30 Archer / 15 Pugilist
Summoner = 30 Arcanist / 15 Thaumaturge
Scholar = 30 Arcanist / 15 Conjurer
Dragoon = 30 Lancer / 15 Marauder
Black Mage = 30 Thaumaturge / 15 Archer Proof
White Mage = 30 Conjurer / 15 Arcanist
Monk = 30 Pugilist / 15 Lancer
Warrior = 30 Marauder / 15 Gladiator
Paladin = 30 Gladiator / 15 Conjurer
So let's say I just got thaumaturge (THM) to 30. To be able to get the quest to change to BLM, I would need to level archer (ARC) to 15 first. Kind of similar to subjobs. After I have done that, I can go to the THM guild and become a BLM.
BLM specializes in direct magic damage spells more so than THM while THM retains many nice cross class abilities.
For instance, THM has access to raise, protect, stoneskin, mantra, etc while BLM does not. In exchange, BLM obtains some nice abilities of its own that THM does not have access to: convert, freeze, apocatastasis, manawall, and flare.
http://www.ffxiah.com/player/Bismarck/Hitome FFXIV: ARR
Hitome Mehan - Midgardsormr
Hito Yu - Midgardsormr