In my 3d scene view, with display mode set to "textured", my texture looks fine. But when I render it the texture stretches all over the place. All my settings are at default, just a single sun object for lighting.

In my 3d scene view, with display mode set to "textured", my texture looks fine. But when I render it the texture stretches all over the place. All my settings are at default, just a single sun object for lighting.

Ensure that you have UV mapping selected in Properties > Textures > Mapping > Coordinates:

If you only have one UV map it's not necessary to select it in the Map drop down, but you can if you want.