Finish is a tasting term. A common word used by wine tasters, it refers to the taste that lingers on the palate after you've swallowed, or spit, a sip of wine.
In wine, one of the factors in determining complexity is the length and depth of the finish. The same is true for cigars. Take a puff, swirl the smoke around in your mouth, and blow it out.
Concentrate on the taste in your mouth after the smoke is gone. Sure, some mild cigars aren't going to have much finish, either in terms of length or complexity. But stronger, more full-bodied cigars will have distinctive flavors that linger for quite a while.