Here's a good explanation that I found:

"What does an EIR do?" - http://framethink.wordpress.com/2009/08/08/what-does-an-eir-...

Ah, nice post. Also, worthwhile: http://framethink.wordpress.com/2009/08/08/many-benefits-and...

Slightly comparable risk to Chris Dixon's why you don't want to raise seed money from a VC firm: http://cdixon.org/2009/08/14/the-problem-with-taking-seed-mo...

We had an EIR when I was in grad school. He would basically just give a lot of advise, and explain why everyone's ideas wouldn't work. The exciting thing is that you potentially hear a lot of nearly-good ideas and you get to help transform them into good ideas.

It depends on each firm, but it usually goes something like this:

You just left a company like Google in a prime role or sold your last startup. You don't know what you want to start yet, but you jive with the firm. They want first dibs on what you do next and/or you already have a great relationship with the firm. It might be joining another startup that comes in through the door or doing your own thing.

Bret Taylor did this with Benchmark before starting FriendFeed. Not sure if he frequents here,but he could give some insight. Jason Calacanis did it with Sequoia as an "Entrepreneur In Action" as well.