All you can do with 10.6.8 is set up your iCloud email account using these settings: http://support.apple.com/kb/HT4864, and possibly your iCloud calendar using this unsupported approach (do this at your own risk): http://hints.macworld.com/article.php?story=20111014102515403.
Setting up iCloud on your Mac requires OS X Lion (10.7.2) or higher. You will need to upgrade to either Lion or Mountain Lion in order to set up iCloud on your Mac. First check to see if your system meets the system requirements to upgrade.
Lion system requirements are:
- Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7, or Xeon processor
- 2GB of memory
- OS X v10.6.6 or later (v10.6.8 recommended)
- 7GB of available space
Mountain Lion requirements are shown here: http://support.apple.com/kb/HT5444.
If you can run Mountain Lion, you can purchase an upgrade from the Mac App Store. If you can only run Lion, contact the online store in your country and purchase a download code to download it from the Mac App Store.
After upgrading, you will find iCloud in System Preferences>iCloud on your Mac, and can set it up as explained here: http://www.apple.com/icloud/setup/mac.html.
Before upgrading, you may want to check the compatibility of your existing programs by checking here: http://roaringapps.com/apps:table.