This is
Pilea peperomioides, the Chinese Money Plant. It's an odd sort of plant in that it appears to be all over the place but never shows up in plant books, stores or botanical gardens. We had some in the nursery where I used to work, and when I finally got an ID for it, it was only accidentally.
I got a good giggle out of the story at the link at the bottom of the Wiki article.
As far as keeping it alive, well, since it doesn't seem to be in any plant books, and I'm too lazy to google (or rather, too tired), I can't give you any "official" care instructions but to me, it looks like your plant is kept in a too dark place, possibly also too wet and in a too-big pot.
What I'd do is, take the plant out of its pot, gently shake off as much soil as you can without damaging the roots, then repot it into fresh soil in a smaller pot. Move it to a bright spot without full sun - and I mean the brightest you can get without exposing it to direct sun for several hours a day. Keep in mind, what seems bright to us, doesn't necessarily seem bright to a plant. Water it carefully, only when the soil has dried out to an inch or so below the surface.