And before we go any further, let's clear one thing up right now: rights are about how a publisher can use your work. It isn't the same thing as payment or purchase. Rights are not about copyright, either. The copyright to your writing always - always - remains yours.
First Rights is a term that when you understand the definition, makes perfect common sense. The first time you publish your writing in any format (including blogs, e-zines, and tiny defunct journals), first rights have been used. Basically, first rights is the right to publish your writing for the first time.
This definition goes into what you can sell as an unpublished story/article/poem. And it's causing some confusion in the publishing world. As more and more writers get their start online, writing for digital publishers or self-publishing their work to their own websites, the line between "published" and "unpublished" becomes ever more gray. Honesty being the best policy, if you have published your writing on a website and want to submit it for payment with someone else, don't claim it as an unpublished work. Instead, note how the writing was used, whether or not it's still viewable online, and (if possible) how many hits it actually received while online.
Whether or not you are paid when someone publishes your work for the first time, you have given them First Rights to your work.
This is huge - many writers feel that first rights to their work is much more valuable than any reprint rights (as do many publishers!) so you need to really consider what your writing is worth before submitting it to non-paying or low-paying markets.