Casual refers to playing Magic: The Gathering without an attempt to enter a DCI-sanctioned tournament. Casual players, making up a significant majority of the customer base for the game, are generally more concerned about having fun than winning. Casual games are often played with theme decks that possess a unifying idea, image, or motif, repeated or developed throughout.
Casual formats or variants are mostly invented by players. Some of them like Commander were endorsed by Wizards of the Coast and received official rules and ban lists. Others like Conspiracy Draft were conceived by Wizards to mix up the gaming experience. And some formats are sanctioned for Magic Online but people started playing these formats casually with real cards, i.e. Pauper Magic. Like the DCI sanctioned formats, most casual variants are Constructed, Limited and Multiplayer formats. 
There are a few Casual Variants which are described in the comprehensive rules: