I'm trying to render a scene in Blender, and as of yesterday (it was not doing this when I was working on it 2-3 days ago), I am seeing orange squares dancing around the rendered image. Google tells me that these are render tiles (I think), but what are they, why are they there, and how do I get rid of them?
They are just indicating the currently rendering tiles:
AFAIK there is no way to turn them off, other than disabling tiled rendering (anyway, IMO they are quite useful).
To disable tiled rendering in cycles, enable progressive refine in Render settings > performance:
To disable tiled rendering in BI, make the tile size at least as large as your render:
Note that disabling tiled rendering will increase render times, as in most cases rendering many small images is faster than rendering one large image. See Are there any advantages of bigger/smaller tiles.