An alternate is something or someone that serves in place of another. An alternative is a second option that does not replace the first. For example, when a road undergoing maintenance is closed to traffic, you have to take an alternate route. But when an under-construction road is still accessible to traffic, you might choose to take an alternative route to avoid congestion. The first option is still there, and the alternative gives you a choice.
The words are also adjectives. As an adjective, alternate means (1) happening in turns, or (2) serving in place of another. Alternative means (1) providing a choice between two or more things, or (2) existing outside the mainstream.
Of course, alternate also works as a verb meaning to take turns. Alternative has no verb sense.
Here are a few examples of the words used as nouns:
The producers … are planning to tap the cast member Matthew James Thomas to serve as an alternate for the leading man. [NY Times]
There is no medium-term alternative to the dollar for the international monetary system. [Reuters]
Portman portrays Nina Sayres, prima ballerina, with a deer-in-the-headlights look, while Mila Kunis is her alternate, seductive and potentially lethal Lily. [Waffle Reviews]
The Motorola Droid 2 Global is a solid Android smartphone for globe-trotting executives looking for a BlackBerry alternative. [CNET]
And here they are adjectives:
For those of you who use this route, signs will be up to direct you to alternate routes. [News 12]
No alternative energy source currently in development is near ready for prime time. [Slate]