Click on the Apple icon.. Top left in Finder...
> About this Mac > Version...what does it say?
You have posted in the Leopard OS 10.5 Forum... If this is indeed your OS... Read on... If not... post back.
The first step in Upgrading... is to Snow Leopard = OS X 10.6.x
It is Not available as a download... It is a Paid Upgrade.
Do this first...
Check that your Mac meets the System Requirements for Snow Leopard...
Snow Leopard Tech Specs
http://support.apple.com/kb/SP575
If so... Purchase a Snow Leopard Install Disc...
http://store.apple.com/us/product/MC573Z/A/mac-os-x-106-snow-leopard
Other countries...
http://support.apple.com/kb/HE57
After the Successful Install, run Software Update to get the latest updates for Snow Leopard.
Be sure to make a Backup of your Current System Before Upgrading...
Next... Check that your Mac meets the Requirements for Mountain Lion.
Upgrade to Mountain Lion
With Regard to ML... Check here for compatibility of 3rd party Software you may be using... Also note that Rosetta is no longer supported in Lion and Mountain Lion...
http://roaringapps.com/apps:table
You should also consider more RAM... Get the Maximum you can for your Mac...
It is important to get the Correct and Matching RAM
See Here > OWC RAM > http://www.macsales.com
The above site also has videos on how to Install RAM should you need it...