Take a look at the InputForm. It will be pretty obvious. You need to "tag" the edges so edges with identical vertices aren't confused.
Graph[{Style[1 \!\(\*OverscriptBox[\(\[DirectedEdge]\), \(1\)]\) 1, Blue],
1 \!\(\*OverscriptBox[\(\[DirectedEdge]\), \(2\)]\) 1,
Style[1 \!\(\*OverscriptBox[\(\[DirectedEdge]\), \(3\)]\) 1, Red],
Style[1 \!\(\*OverscriptBox[\(\[DirectedEdge]\), \(4\)]\) 1, Red, Dashed]}]