Actually, the airport express can be used without a router. That's the (part of) the whole point. I suppose it's a bit of a swiss army knife device, so it's not correct to say "whole point" - different strokes for different folks. I think its versatility is what confuses some people and likely leads to a few scratched heads and lost sales.
So, without a router, you can still use it for music streaming, a print server as well as wireless access to a broadband connection (connect the modem to its ethernet port). Works in a hotel as well, just connect it to the ethernet jack in the room.
And yes, I've heard you can use it as an extender for many other wireless products - though I don't have a link saved and I can't remember which specific products. Someone else makes the components in the guts of the Apple boxes. The reason why extenders aren't universal is because everyone's doing a bit of their own magic.
Bruno