These three words are very close in form and have related but not identical meanings. There is also a British English / American English aspect, something I don’t often cover in these Troublesome Pairs, but one that needs addressing here.
Alternative means available as another possibility, or mutually exclusive, said of one of more things (originally, it could only be used of two entities, but matters have freed up and more than two alternatives are now allowed). So you could use the services of one typist or an alternative typist if the first one is busy, or you could choose between the alternative answers “yes” or “no” when answering a question. Alternatives are the two or more available possibilities.
Alternative can also mean activities that depart from and/or challenge traditional norms – like alternative health and alternative lifestyles – the same meaning is there, in that they are offering an available possibility other than the “main” one.
The adjective alternate (with the stress on the middle syllable: alTERnate) means every other – “We hold the event in alternate months” means the event is held in, for example, January, March, May, etc. In biology, leaves alternating on a stem will appear on each side in turn, rather than being in matched pairs, two leaves together. But it’s worth noting that the American English usage of alternate carries the same meaning as the British English alternative defined above. To alternate, the verb (with the stress on the first syllable: ALternate) means to swap between two contrasting options repeatedly – “I alternated between watching the diving and hiding behind a cushion”. If two things are alternating then they are doing the swapping – “Mary and Sue were alternating as the front runners in the race”.
You can find more troublesome pairs here and the index to them all so far is here.