Most 2D games don't use engines. Usually they write their own game-specific logic code, because 2D games vary alot more than 3D games do.
So 2D games usually use APIs - one for sound, one for graphics, and so on. But then the programmer uses the API to write their own " "engine" " code for their game - they write their own map loading/saving/drawing code, movement code, etc...
While 2D game engines do exist, usually they are either very generic (so you'd be just as well off writing your own code), or very specific (so unless you are making exactly the type of game the engine author had in mind, you're out of luck).
Are you looking for a set of tools for easily making maps? That's a different subject. Try Tiled. It's not an engine, but is a tiled map editor.
Many 3D game engines like Unity now offer some basic 2D features as well. See my earlier comment about being so generic that you are often better served to write it yourself.
If you don't know how to program, you can check out RPG Maker (see my earlier comment about being too specific)... If you already know C# and are an experienced programmer, I'd suggest finding some good graphic APIs and get crackin' at your own code.
For 3D games, engines are almost a necessity. For 2D games, they can almost be a hindrance.
