To avail yourself of something means to take advantage of an opportunity or resource that's available to you. When your teacher says students can avail themselves of extra help after school, she means you can use that opportunity if you need it. When a library announces new computers that patrons may avail themselves of, they're inviting people to make use of those computers.

The word often appears in more formal contexts, which is why you're more likely to see it in writing than hear it in everyday conversation. A sign might read “Please avail yourself of the refreshments,” meaning “Help yourself to the snacks and drinks.”

You'll also encounter avail in the phrase “to no avail,” which means without success or without producing the desired result. If you search everywhere for your missing homework to no avail, you never find it. If a soccer team practices new plays but loses the championship, it was to no avail, meaning their extra effort didn't change the outcome.

The related word available is much more common: it simply means something can be obtained or used. When your friend is available to play after school, they're free. When a book is available at the library, you can check it out.