Yes it is possible. There are many, many existing patents related to games played on standalone computers and on networked computers.
Community poker card game online playing system
US 8388428 B
Mar 5, 2013
If you go to google patents and search for "game internet" and restrict the search to granted patents and to U.S. patents you will see many more examples.
You ask "what kind of patent?" Between design patents, utility patents and the somewhat obscure plant patents, the only choice for something functional is a utility patent. Within a utility patent there can be method claims to the steps involved, system claims, apparatus claims and claims to the recorded media with the software on it.
Avoiding your claims being seen now as merely an abstract idea takes quite a bit of expertise in patent drafting. Avoiding them being seen as abstract by judges in the future requires a crystal ball. A case pending now at the Supreme Court may clear it up somewhat.