Both words can be use adjectives and come from the Latin word alter, meaning "other."
Alternate means "set up or following by turns, one after the other" or "a substitute or second" or "every other in a series." The adverb form is alternately.
Example: The captains alternately chose players for their respective teams. (Following by turns)
Alternative refers to or implies a choice between two things. As a noun, it refers to the other choice. The adverb form is alternatively.
Example: If the flight is grounded, our alternative is to travel by train. (The other choice)
Alternate, with the accent on the last syllable, can be a verb. It is related to the adjective and means "to go back and forth" or "to change from one state to another and back again repeatedly."