From Diablo Wiki
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). 
- 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.
- Bounties are not repeatable in the same game. You need to leave and begin a new game to refresh the bounties available.
Information below this point has been datamined from the Reaper of Souls PTR game files and as such should be considered a spoiler.
 Datamining Bounties
In one of the panels of images datamined called 'WaypointMap' are icons now known to be bounty indicators. All the images on the panel are in pairs, one plain and one with a green tick over it, probably indicating completion.
It does not necessarily mean that all of the icons represent a type of Bounty. All the icons are below.
|Bounty with a random boss, and the completed icon.|
|Bounty to complete an event, and the completed icon.|
|Bounty to clear a dungeon, and the completed icon.|
|Bounty with a quest or act boss.|
|This could be something to do with the Devil's Hand as the icons are similar to the image panel labelled Devil's Hand in the data files|
|This marks a completed bounty in a given act. When all five bounties are completed in a given act, a special random reward can be earned by visiting Tyrael in town.|