Hey there! Welcome to the forum!
[img]https://t00.deviantart.net/GYzqKLTx87KNT4wxcugPeKGXNVk=/300x200/filters:fixed_height(100,100):origin()/pre00/c661/th/pre/f/2012/247/2/7/chikorita_by_kizarin-d5djrzk.png[/img]

Free RP: Basic one liners or so. Think chat-based RP, like instant messanger. 
Casual RP: Paragraph or a few (2-3). Average depth, perfect blend between free and advanced that a lot of people like here.
Advanced RP: Paragraphs usually depend. Some people have a ton of paragraphs, some don't. However, these RP have a lot of depth and background to them.
Arena RP: 1x1 (or more) combat fights between RPers.
1x1 RP: Like it says. One RPer with another.
Nation RP: Control a nation, do diplomatic stuff or fight battles.
Tabletop RP: Think dungeons and dragons and you'll understand what tabletop is.

I hope this gets you at least started on your path here. If you have any more questions, don't be afraid to ask. Otherwise, Poi out.
[img]http://static.tvtropes.org/pmwiki/pub/images/ChikoritaReaction_1415.png[/img]