Basics
If you're looking for the FAQ, it is located here. Beta and Gamma descriptions taken from there.
Beta
The job of a Beta is to proof read a story for any spelling/grammatical errors. They're the second person to read what you've written and they're basically an editor.
They will go over your work(s) and tell you how you can improve. They will also help to keep your story focused, keep the characters in character, help you with character development, plot development, and help smooth out ideas.. Betas are supposed to give you an honest critic, and point out any major flaws.
Many have strengths in certain areas. It's helpful to find someone who has a strength in your weak area.
Gamma
Gammas are people who would like to have their written works read over by another in order to check for spelling/grammar mistakes.
They are the original author who is seeking someone to help them edit.
Fanfiction
Fan fiction is taking someone else's characters or settings and using them in your own story. Even if you use your own characters, using the elements of another's work in your story makes it fan fiction. For example, Vampire: the Masquerade. There are no canon characters, but if you're using their settings for you story, then it's still considered fan fiction.
Original fiction/work
The characters and elements have been created by you for your story.
