Some uses for the Internet Archive's Wayback Machine:
When you come across a broken link, check for an archived version.
Example: Greenpeace blog post from 2008, no longer live on the web
Archived version (crawled on March 18, 2012): https://web.archive.org/web/20120318201527/http://weblog.greenpeace.org/makingwaves/archives/2008/10/coal_gets_the_boot_in_the_gree.html
Consider checking multiple versions (if the site has been crawled more than once)
To see what a website looked like at a specific point in time
It's easiest to find archived websites when you already have a specific site in mind. Using a different search engine can help you discover relevant sites, both current and older.