A first reader is just what you might suspect: he or she is one of the first people to read and comment on a new piece of writing before the author does final revisions and submits it for publication.
Very few writers are at a place in their craft and career where they can write in complete isolation, finish a story, send it off to an editor, and have that work see print. The vast majority of us need feedback on our work before it's ready for an editor's eyes.
When you're just starting out as a writer, a good critique group is enormously helpful. But if the group is really good, your workshop partners will also be busy writing and submitting their stories ... and once you start seriously working as a writer, keeping up with everyone else's creative output can become a challenge. I've known a lot of pros who've had to drop out of productive critique groups simply due to time pressures.
Consequently, some pros who are in the middle of working on sold-on-proposal books under contracts with big publishers come to rely mainly on their editors for feedback. Which by some lights is entirely sensible: your editor is the most important person you need to please before the book or story goes to press.
But in the grand scheme, your editor may not be enough. Any good editor can give you excellent advice about fiction basics: plot tension, characterization, dialog. All hugely important. But at large houses you may find yourself assigned to an editor who may not have read widely in the particular genre you're writing in, and that otherwise excellent editor just won't realize when elements of your story veer too closely to works by other authors in the genre. That excellent editor won't be able to say "Hey, Author X did something like this in That Book I Just Read ... why don't you try this other thing instead?"
So, it's always a good idea to have other people read your works in progress. But what do you do when you're just too busy making a living as a writer to participate in a critique group?
You cultivate a group of first readers (or beta readers, if you prefer terminology borrowed from the bustling world of fan fiction).
Critique partners are always peers: fellow writers. They'll have opinions about the mechanics of your story or novel, and they may not be fans of the genre you're writing in. Critique partners are expected to be tough and honest, and they'll be coming at you from the perspective that your work ... well, it needs more work.
Your first readers, on the other hand, need not be fellow writers. They don't need to have strong opinions about the mechanics of a story or chapter -- you're a working professional writer, remember? You know how to fix this stuff on your own -- but you do need to be able to rely on them to tell you honestly when something isn't working. Their job doesn't have to be to tell you how to fix things, but you need to know that they will unfailingly draw your attention to problems. Therefore, your first readers need to be well-read in the genres you're writing in, and they need to have excellent instincts about what makes for a good story or novel.
Furthermore, they need to be fans of your work. Not in terms of thinking you're infallible as a writer, but in terms of them fundamentally getting what it is you're trying to accomplish creatively and being excited at the prospect of helping you get there. This is crucial. You're done with critique partners who can hardly hide their boredom at reading horror when they'd rather be reading literature, but with a heavy sigh they'll read your stuff because you read theirs. You have a deadline, and you need people who are enthusiastic about what you're doing, and who are willing to read your pages and give you feedback when you need it, not when the next meeting is scheduled.
Where do you find good first readers? Sometimes you'll find enthusiastic peers in the critique groups you've been involved with; it's simple courtesy to return the favor and critique their work, but typically they'll understand the whole deadline thing and they'll ask for your help when you're better able to give it. Other times, you can run into good potential first readers at conventions, or recruit well-read acquaintances.
But whatever you do, once you've recruited good first readers, treat them right. They're the best friends your fiction will have. Acknowledge them in your books, and make sure they get first dibs on your authors' copies. Gift certificates and other more substantial thank-yous don't hurt, either.