I have a iMac w/ MAC OS x10.5.8 and I need to upgrade to os 10.6.8 or better. Can I do that on a iMac?

Start by checking if you can run Snow Leopard:

Requirements for OS X 10.6 'Snow Leopard'

http://support.apple.com/kb/SP575

OS 10.6 Snow Leopard is once again available from the Apple Store:

http://store.apple.com/us/product/MC573/mac-os-x-106-snow-leopard

and in the UK:

http://store.apple.com/uk/product/MC573/mac-os-x-106-snow-leopard

but nobody knows for how long it will be available.

To use iCloud you have to upgrade at least to Lion, but some functions are only available in Mountain Lion:

http://support.apple.com/kb/HT4759

You can also purchase the code to use to download Lion (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:

http://www.apple.com/osx/specs/

When you have installed it, run Software Update to download and install the latest updates for Snow Leopard.

Upgrade Paths to Snow Leopard, Lion, and/or Mountain Lion

You can upgrade to Mountain Lion from Lion or directly from Snow Leopard. Mountain Lion can be downloaded from the Mac App Store for $19.99. To access the App Store you must have Snow Leopard 10.6.6 or later installed.

Upgrading to Snow Leopard

You must purchase Snow Leopard through the Apple Store: Mac OS X 10.6 Snow Leopard - Apple Store (U.S.). The price is $19.99 plus tax. You will be sent physical media by mail after placing your order.

After you install Snow Leopard you will have to download and install the Mac OS X 10.6.8 Update Combo v1.1 to update Snow Leopard to 10.6.8 and give you access to the App Store. Access to the App Store enables you to download Mountain Lion if your computer meets the requirements.

Snow Leopard General Requirements

1. Mac computer with an Intel processor

2. 1GB of memory

3. 5GB of available disk space

4. DVD drive for installation

5. Some features require a compatible Internet service provider;

fees may apply.

6. Some features require Apple’s MobileMe service; fees and

terms apply.

Upgrading to Lion

If your computer does not meet the requirements to install Mountain Lion, it may still meet the requirements to install Lion.

You can purchase Lion by contacting Customer Service: Contacting Apple for support and service - this includes international calling numbers. The cost is $19.99 (as it was before) plus tax. It's a download. You will get an email containing a redemption code that you then use at the Mac App Store to download Lion. Save a copy of that installer to your Downloads folder because the installer deletes itself at the end of the installation.

Lion System Requirements

1. Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7,

or Xeon processor

2. 2GB of memory

3. OS X v10.6.6 or later (v10.6.8 recommended)

4. 7GB of available space

5. Some features require an Apple ID; terms apply.

Upgrading to Mountain Lion

To upgrade to Mountain Lion you must have Snow Leopard 10.6.8 or Lion installed. Purchase and download Mountain Lion from the App Store. Sign in using your Apple ID. Mountain Lion is $19.99 plus tax. The file is quite large, over 4 GBs, so allow some time to download. It would be preferable to use Ethernet because it is nearly four times faster than wireless.

OS X Mountain Lion - System Requirements

Macs that can be upgraded to OS X Mountain Lion

1. iMac (Mid 2007 or newer)

2. MacBook (Late 2008 Aluminum, or Early 2009 or newer)

3. MacBook Pro (Mid/Late 2007 or newer)

4. MacBook Air (Late 2008 or newer)

5. Mac mini (Early 2009 or newer)

6. Mac Pro (Early 2008 or newer)

7. Xserve (Early 2009)

Are my applications compatible?

See App Compatibility Table - RoaringApps.

Am I eligible for the free upgrade?

See Apple - Free OS X Mountain Lion upgrade Program.

For a complete How-To introduction from Apple see Upgrade to OS X Mountain Lion.