You have Mac OS X Leopard, and an Early 2009 MacBook is compatible with OS X Mountain Lion. First, as you have an Intel-based Mac, you can upgrade to Snow Leopard, so buy Snow Leopard.
Then, make a backup of your files, insert the DVD and upgrade to Snow Leopard. After upgrading, open > Software Update to update to 10.6.8, so App Store will be installed on your computer.
After that, check if your Mac is compatible with OS X Mountain Lion > http://www.apple.com/osx/specs. Also, see if your applications are supported > http://www.roaringapps.com. If everything is compatible, open App Store and purchase Mountain Lion. OS X Mountain Lion will be downloaded and the installer will show up when it finishes, so follow the steps to upgrade to Mountain Lion.
Apple says that you can run OS X Mountain Lion with 2 GB of memory, but I recommend to install 4 GB of memory or more to guarantee the best performance. You can buy memory in OWC or Crucial > http://eshop.macsales.com/shop/memory/Apple_MacBook/White/2.13GHz