As an adjective, “alternate” means “every other,” and it is usually used in relation to time or objects:
We were asked to focus on alternate lines of the figure. (Every other one.)
“Alternate” is also a verb, meaning “to switch back and forth in turns”:
The wet season alternates with the dry season.
“Alternative” denotes that a choice was made between at least two things:
He chose the polygon method as the best alternative for measuring compressible subsonic flow.