I know on the SAT the essay is incorporated into the Writing score. Does the ACT do something similar? Or is the essay just a separate score that's not included in your composite at all?
No, your ACT essay score will not have any effect on your composite score. There is a score on the ACT score report called "Combined English and Writing" that, as you might expect, combines your scores on the English section and the Writing section into a single score out of 36. However, this score isn't incorporated into your composite score. The composite depends exclusively on the four multiple choice sections of the ACT. Colleges will look at your Writing score separately if they're interested in it.
Also, you might be interested to know that the SAT is changing its essay policy with the new version of the test. The essay will be optional, and it won't be incorporated into your total score (basically the same deal as the ACT).