I like atzy's られる explanation is very good. However, I believe that your question is about something that we can/cannot eat. If so, 食べられる is a correct way of saying YOU CAN EAT IT. 食べれる is grammartically incorrect, many people started using it because it's easy to say it. I will never use it forever because I feel ashamed of using it that way.
As a student who just took a vernacular Japanese class I can answer this. Simple answer, 食べれる is vernacular Japanese, used when speaking. 食べられる is standard Japanese, i.e. The Japanese that is taught in school!
The rule for this is you can take any group II/る verb and drop the ら from the -られる to make it れる。