Q: What are the tiny flies in my potting soil?

and how do I deal with them?

Turns out these are what are known as fungus gnats. I have them in my potted plants in the house, occasionally; turns out they thrive in an environment where there is plenty of moisture, such as an overwatered houseplant. The larvae live in the soil and feed on fungi and plant roots, and the adults swarm around the surface until they get a chance to breed again. 

The solution is pretty simple: don't water your plants so much. Also don't let them stand in water in a pot that's not draining or drying out. I had a new plant that was the focus of the latest infestation, so I stopped watering it until the soil had dried down from the surface about an inch or two. I also put the plant outside briefly each day, coming back 5 or 10 minutes later to blow away some of the adult insects from the surface and take the plant back inside real quick before they could find the soil again.

There are insecticides for these bugs but the best thing to do is really just let the topsoil dry out. You can also put a layer of sand on the top of the soil (keeps the larvae under the sand, and the adults out of the soil), but who's got room in their indoor plants for a couple of inches of sand? Not me.

edit 2021: I have also used "sticky stakes" to trap the gnats that do end up in the house, and they are very effective combined with the methods listed above.