What is feedback?
Feedback is telling an author what you thought of her fanfic. Feedback can simply be telling an author you loved her fic but can also consist of constructive criticism.
Every author loves feedback. Hearing that people are reading and enjoying what you wrote is amazing. However, most authors would prefer feedback that is deeper than "Oooh, I loved it! Please write more!" Although that type of feedback is wonderful (especially for the author's ego), the point of feedback is to help the author become a better writer.
How can you help someone become a better writer?
Easy. Tell the author what you enjoyed about the fic. Do you feel she portrayed a character especially well, or that her plot was unique? Tell her. But also tell the author what you feel she could work on. Is the grammar and punctuation of a piece lacking? Suggest a beta reader. Do you feel that a particular sentence is awkward? Let her know and perhaps even suggest a better way to write it.
Don't say: "Your grammar sucks and you can't write to save your life." That's NOT feedback, that's flaming. Flaming is giving unconstructive, hurtful criticism.
Do say: "I really enjoyed your fic. I think I saw a few grammatical errors, so you might want to check for those. However, your characterization of Willow was great!"
Beginning and ending with praise or something you liked while discussing what you feel could be improved in the middle is a good way to give feedback. It tells the author both what you enjoyed and what you felt needed work in a constructive and polite manner.
Why should I send feedback?
Authors work extremely hard on their fics. With each posting of a fic, they expose who they are to the world. If you read the fic, then let the author know what you think! It is the only pay a fanfic author receives. Even if the feedback is mostly constructive criticism, a majority of authors will be pleased to know someone is reading their fic. It is disconcerting and disheartening to post something to a list and receive no feedback. It makes an author wonder if everyone is just deleting without reading. If an author doesn't receive feedback, it is possible she will stop writing altogether. After all, if "nobody" reads it why should she work that hard?
So send feedback today! Even those authors you feel already know how amazing they are will be gratified and pleased to hear from you!
I'm an author. What should I do with the feedback I get?
Respond to the feedback you receive. A simple thank you suffices, though commenting on what the reader said is always good. It makes it more likely that they will feedback to you again and keep reading your fics! I know I have personally stopped feedbacking to a few authors because I received no response. Why should anyone take the time to tell you what they thought if you can't acknowledge that they put energy into giving you an intelligent, well-thought out email about your fic?
Both sides have responsibilities concerning feedback. Don't forget yours!
(For purposes of this essay, I assumed the author would be a woman. Why? Because 90% of fanfic writers are women. *grins* But that doesn't mean there aren't talented male writers either!)