When you're facing a fork in the road, you have two possibilities, or alternatives. You could take the left fork and see where it leads, or you could choose the path that goes to the right.
Whenever there is an alternative, it involves a choice. A wealthy student has the choice of going to a public school, or the alternative, a private school. Alternative can also refer to a different way of doing things, outside the norm. You might say that a hermit who lives in remote a cave with a pet turtle is living an alternative lifestyle.