How is the real world value of bitcoins generated?
BTC works as any currency. As long as people see value in it, it has value.

The momentum was just the curiosity the users had... Also the whole "Make free money with your computer"- thing which was started.

A big influence was the famous "How does bitcoins work" video I saw on vimeo. That got ME started at least. (Link to updated version)

The algorithms your computer solves when "mining" isn't used for anything no.
There are also other kinds of crypto currencies on the rise. The most notable is Litecoin. It uses a different algorithm. (Scrypt, instead of BTC's double sha256). The point of Litecoin was to make users able to "mine" more efficiently on their CPU's.

I hope this somewhat answered your question.