This would be very easy with DBMan. I would suggest visiting JPDeni's DBMan tutorial to get started.

Using the referenced signup form you provided above you can easily use the same fields for your database.

JPDeni's site contains a configurator which will guide you through setting up your database fields and initial forms which can then be modified to suit your needs.

http://jpdeni.com/dbman/

There is also a DBMan FAQ which provides many tips and tricks which is noted below.

Unoffical DBMan FAQ

http://creativecomputingweb.com/dbman/index.shtml/