This is an old southern term used by my grandmother. It is a disrespectful term for a white person similar to "cracker" or "honky." I believe it comes from the fact that woodpeckers have reddish hair up toplikewhite people.
To give someone who likes you just enough hope to keep them coming without the relationship actually going anywhere or advancing, and it usually keeps the victim in emotional confusion and pain.
Friend: Megan is taking days to respond to my texts, but she also seems to like me; things aren't adding up bro.
Me: I think she is peekabooing you.