If you have Mac OS X 10.5.8, you have Leopard, not Snow Leopard which is Mac OS X 10.6. The most recent version of iTunes, iTunes 10.7, requires Mac OS X 10.6.8 Snow Leopard which is not a free upgrade. You can try iTunes 10.6.3:
That's worked for some people who updated their iOS device to iOS 6. If not, then you'll need to upgrade your Mac to Mac OS X 10.6 (still available if you call the Apple Store, but you must have an Intel-based Mac) so you can run iTunes 10.7.
Start by checking if you can run Snow Leopard:
Requirements for OS X 10.6 'Snow Leopard'
Whilst Apple have withdrawn Snow Leopard from their stores, you can still get it from Apple by calling 1-800-MY-APPLE (if you are in the USA) and they will supply the SL DVD for $20 for a single user, or $30 for a family pack that covers up to 5 Macs. You can also purchase the code to use to download Lion from the same number (Lion requires an Intel-based Mac with a Core 2 Duo, i3, i5, i7 or Xeon processor and 2GB of RAM, running the latest version of Snow Leopard), or you can purchase Mountain Lion from the App Store - if you can run that:
If you are outside the US call your national Apple Helpline:
If you're in the UK, use this number: 0871 508 4400
When you have installed it, run Software Update to download and install the latest updates for Snow Leopard.
To use iCloud you have to upgrade all the way to Mountain Lion: