I suspect you will be hard pressed to find any gaming apps that won't mirror to the Apple TV, but many content providers will block their apps and content streamed for viewing through web browsers, outside of media I suspect you will find little that can't be mirrored.
There are 2 types of AirPlay, basic and mirroring. Basic AirPlay is far more reliable since it isn't a live representation of your screen, Many apps/content may not work with basic AirPlay since the developer needs to specifically opt in and enable basic AirPlay. MIrroring is less reliable since it's live (although that being said there is invariabally a slight delay which makes gaming difficult in many circumstances), but is on by default unless the developer or provider specifically opts out by disabling mirroring.