Alternate:

• to occur in turn; to interchange regularly or in succession
• occurring by turns; every other or every second something in a series

Alternative:

• a possible choice between two or more things
• the presenting of a choice between two or more possibilities
• of two things that are mutually exclusive
• activities (e.g. lifestyle) that are not considered conventional or traditional

In American English, alternate has come to be used in the sense of something being another possibility or choice. (The Chicago Manual of Style suggests this is appropriate when that something can be regarded as a substitute.) It is not common to see this in British English, and it is often regarded as incorrect.