Bounties are set tasks. For example, locate and kill a Superunique or Boss monster or complete an event or clear a dungeon and be rewarded with Experience, Gold and possibly a Rift Keystone. You access a Bounty by using the Waypoint map and selecting a waypoint marked with an exclamation mark (!) or Diablo's skull icon (indicating boss or quest boss). ^[1]

There are 25 random bounties per game, 5 per act.
Known Bounty types are "Complete an Event", "Clear a Dungeon" and "Kill monster/boss".
You do not need to complete Campaign mode to access Adventure mode and Bounties.
There is no difficulty mode such as normal, nightmare etc. The difficulty is scaled to your character level. It's unknown what happens in multiplayer games. It could take the average character level or just the level of the person who created the game.
Rewards for completing Bounties can sometimes be a Rift Keystone which are used to open Nephalem Rifts.

Gallery

Game Settings
Adventure mode Waypoint maps

References

↑ Bounty Details - IncGamers' Gameplay Report, 08/11/13

[bountydetails-1] Bounty Details - IncGamers' Gameplay Report, 08/11/13

[1]