1st you will need to modify the firmware of your 360 drive so that it will allow you to lay backups - thankfully this is easily done on 99.9% of drives
Full tutorials and all software tools required are available on this site.
2nd, if you want to backup your OWN games then you will need to buy a Samsung drive for your PC and reflash the firmware with special KREON Firmware, this allows the drive to read Xbox 360 discs, all software needed to create backup games is available on this site - as is the KREON firmware.
You best and easiest option is to simply download already created backups
You could check Xbox 360 DVD Drive Database for lot number on your box (google it). Otherwise, you might be able to tell from the shape of the cutouts in the dvd tray. If you still stuck then open her up and take a look.
I think there is also a method to make backups by connected your Xbox 360 drive to your pc and dumping onto your hard drive that way. But I have never made my own backups, I used dl'ed ones.
The only definate way of determining what drive you have is by opening the drive tray, but doing this will only tell you the drive manufacturer, it will nt tell you what firmware the drive has - it's the firmware that makes all the difference, most are piss easy to work on, only 1 is a bastard and needs a chip fitted.
The backups are 'kept' on the Internet, I'm not sure what yuo mean to be honest.