A Quick Recommendation
This is no longer relevant! Facebook has changed it yet again, and it’s different in different context so there is no clear answer right now. BIG just upload your photos with 2048 pixels or larger on the long dimension.
Now Updated For The New Size!
Well as near as I can tell everyone is now able to upload and be shown the new Facebook image size. The largest photo you can have on Facebook is now 720x720px. That’s 42% larger (by area) than the old standard of 604x604px. It’s so big that it doesn’t fit into my blog template. Click the image to see how large it actually is.
So if you want to get picky about re-sizing your images for better quality and watching for artifacts you should shoot for 720 pixels either wide, tall or both.
It seems that no matter what you do Facebook re-compresses your images regardless of what their original file size might be, this equates to about a 50% quality setting in most apps but there seems to be no size it won’t re-compress, even if it actually makes the file bigger.
Mike makes a good point, you can find out more in his Facebook Gallery