What causes a circle of light to appear opposite the sun when looking from an airplane? Today I saw a circle of light outside my plane window on the clouds, as if someone was shining a bright, tightly focused flashlight, or perhaps like the halo that sometimes appears around the sun. I think it was approximately where I would expect our shadow to be (at least the sun was shining on the opposite side of the plane).

I took a video with my phone. I move the camera around a bit to show that it doesn't seem to be an artifact of the window.
 A: It is an optical phenomenon called a "glory" caused by a diffraction in very small water drops. In fact, the radius of the glory depends on the size of the drops of water and can change in a dynamic fashion as you fly above different clouds with different size of drops. It is quite different than a rainbow though.
Professor Lewin gave a fantastic lecture at MIT about the rainbow and other optical phenomenon, where he mentions glories: https://www.youtube.com/watch?v=pj0wXRLXai8&list=PLyQSN7X0ro2314mKyUiOILaOC2hk6Pc3j&index=32. 
However, as pointed out in the comments, he does not provide a thorough explanation of the glory which is extremely difficult, there is still no complete explanation that is agreed upon. Still the lecture as a whole is very instructive and is worth watching.
A: What you're seeing is called a glory, an optical phenomenon related to rainbows, caused by reflections and refraction inside the water drops in the cloud (plus some additional physics, which are not fully agreed on; cf. this question). If the cloud had been closer you would have been able to see the shadow of the plane embedded in the glory:

Image source
For more information see the page on glories at Atmospheric Optics, particularly regarding their formation. (Also, just go to Atmospheric Optics and have a browse! they've got all sorts of gorgeous stuff there.)
