The two English words 'alternate' and 'alternative' seem to be a bit confusing to many people. The noun 'alternative' simply means 'something that can be done/used instead of something else' and nobody replaces it with 'alternate'. When it comes to the adjective in the same sense, though, some people - especially those who use American English - consider both as equivalents.
Do we have any alternatives? (Are there any other things that we can do/use instead?)
We have to find an alternative/alternate[esp. AmE] solution.
However, it's to be noted that not everybody considers this use of 'alternate' acceptable. So it may be safer to stick to 'alternative' in such contexts.
The word 'alternate', on the other hand, is used as an adjective to mean that two things follow each other again and again. When talking about many things of the same sort occurring in a row (for example days,nights...), it means 'every other (one but not the next)'.
The fabric had alternate stripes of black and white. (black, white, black, white...)
The fabric had alternate stripes of black and white. (black, white, black, white...)
Meetings are held on alternate week days. (One week day, not the next, but again the third, not the fourth, and so on.)
The word is also used as a verb when two things follow each other repeatedly.
White stripes alternate with black ones on that fabric.
You have to alternate layers of bread and/with cheese when making a cheese sandwich.
The lighting on the stage alternated between dimness and brightness.