Once you reach player level 6, Daily Quests will start to appear at 2 AM PST (10 AM UTC) on North American servers.  This time is not adjusted for daylight savings, so if you live in an area under daylight savings, then quests will appear an hour later.

You can retain up to three Daily Quests at one time.  Unlike Hearthstone, you cannot abandon a quest.  They persist until you complete them.

The [possible Daily Quests][1] are currently:

 - Play 8 Games (Any Mode) - 800 gold
 - Win 3 Games (Any Mode) - 600 gold
 - Play 3 Games as an Assassin Hero - 300 gold
 - Play 3 Games as a Warrior Hero - 300 gold
 - Play 3 Games as a Support Hero - 300 gold
 - Play 3 Games as a Specialist Hero - 300 gold
 - Play 2 Games As a Warcraft Hero - 200 gold
 - Play 2 Games As a Diablo Hero - 200 gold
 - Play 2 Games As a StarCraft Hero - 200 gold

 [1]:http://heroesofthestorm.wikia.com/wiki/Daily_Quest