A push assignment is when, instead of you requesting to review an item, Eden requests you to do it, usually because a manufacturer wants reviews on their items. You'll get an email saying you have an assignment, and you can click through and look at it before deciding whether to accept or not. You can reject the assignment (maybe the product is a cock ring, and you're a single girl; there are ingredients you're allergic to; you just think it's ugly and don't want to do it) and they'll just pick someone else. Saying no doesn't make you less likely to be chosen in the future, and you don't have to say why.
I think anyone who is a regular reviewer is eligible, but I don't know how they decide who to push to. I just got my first push assignment this month, and I think I started reviewing here in September 2011.
Edit: After looking at Nazaress's post, this looks confusing. A push assignment can
be a Descriptive Review, but it might just be a regular review. It's regular reviews that can just randomly appear in your mailbox one day. I'm not in the Descriptive Review program, but I got a Push Assignment this month.