Of course, it was rejected by all the publishers I sent it to.

Then, in January, out of the blue, I
received a postcard from an editor of a very small press company. Most of their work goes into creating on-line magazines but the publisher occasionally puts out printed anthologies. He said he wanted to use my story in his upcoming anthology.

Since then, we have exchanged e-mail addresses, and he still expresses interested in publishing the story, but I have no idea how to proceed.

Can anyone give me advice on what I should be doing, or asking him? What are my rights as the author? What kind of payment should I expect?

Any help would be appreciated.
 

If it were me, I would do it in a heartbeat, because I don't care about getting paid or even getting credited as the author. Since I have to assume that you care about both of those, I don't know what to tell you. Just like the discussion on writing for the Young Adult market, I can't add anything useful.

But for me, the important thing is that this guy values your work. That's the key element in this whole situation. Congratulations
 

What you want early on is as much published work to show up on your resume as possible. You can succeed without it, but it is harder.

Bottom line, get copies as payment and accept the offer graciously!