What's the name of this small connector?
A short name would be an IDC connector, here's a whole list of them: 
http://www.jst.com/home21.html

This one looks close to what you want:

http://www.jst-mfg.com/product/detail_e.php?series=246

Just measure the pitch (pin to pin spacing) and you should be able to find a mate.  If I remember right, finding those in the US at that pitch and in 1-100 unit volumes was a pain.  JST was eventually the guys who had what I was looking for though.