Quests are given by NPCs in settlements. They are separated into 3 categories - Normal, Hard and Nuclear.
The Normal Quests
The normal quests can involve the delivery of a package - is it a weapon, a bomb or a head, you don't care. You deliver it and get paid. It can involve the killing of a monster - a mole rat, a ghoul, a rad-roach. You get paid per monsters killed, capped at 20 monsters per quest.
The Hard Quests
A hard quest can be something like this:
Trace and kill Raider Randomnamehere. Bounty whatever you scavenge from him + 300 caps. 600 if you bring the head.
The raider that needs to be killed can either be a NPC(at lower levels) or an actual person. When it's an actual person, you go to the wasteland and type a lengthy, long RP message describing how you traverse the landscape in the search of the old b*stard. After that, it's the raider's turn to write. You start a fight and if:
The winner is the Wanderer - Raider loses 2 points from 3 skills and 300 caps/weapon/armor, Wanderer, gains 2 points in 3 skills by choice, 300 caps/weapon/armor from raider.
The winner is the Raider - Wanderer loses 2 points from 3 skills and whatever the raider takes from him. The Wanderer is then resurrected in the city he took the quest from and the quest is failed.
That's about it.
Nuclear quests will be given from the factions. They involve the extermination of Deathclaws and/or Mutants. They also involve the duels between soldiers from the 2 factions.