It renders like this, but when I play the game it has no texture. Why doesn't the texture show up in the game engine?
Here is the file: http://filedropper.com/textured
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It seems that you haven't unwrapped the mesh. Change to Edit Mode in the 3D-View and select all A press U for unwrap.
In the texures settings select UV for Coordinates and UVMap as Map, Projection should be Flat by default.
Set the Viewport Shading to Texture
After that the texture should be shown in the Game Engine.
Examining your .blend shows that the texture wasn't included in the file. I replaced it with another one, the issue that the texture isn't rendered in the BGE could be fixed either by moving it into the first slot:
Alternatively you could set Shading to GLSL in Properties Panel N in the 3D-View in Object Mode: