For the most part, I think I have this one figured out, but I still like to double check myself whenever I use either of these words.
Most frequently, affect is a verb, meaning to influence or cause a change in, or to act on the emotions of. "I was very affected by such a sad movie." Effect is usually seen as a noun, meaning something brought about by a cause (i.e., a result), or the power to produce a result. "The sad movie had a profound effect on me."
Of course, just to make things interesting, we can pull a switcheroo with these words, and make affect a noun, meaning feeling or emotion. This is especially used in psychology: "The patient's affect was blunted and reduced," meaning that they showed little emotion. Use effect as a verb, and it means to produce or bring about. "The new manager plans to implement a change in suppliers in order to effect cost savings." (The change in suppliers will produce cost savings. In the sentence I gave as an example, the word "implement" could be replaced with "effect.")
If anyone has an easy way to remember the difference between the two, please share. This is one that took me many years to get straight, and it's still not automatic for me.