Automatic upgrade
PrestaShop introduced automatic upgrade in version 1.4.4. It enables shop owners to backup and update their site in a few clicks and no technical knowledge. It is explained in full details in the User Guide's Tools chapter, in the "Upgrade" section.
If upgrading from a version between v1.3.x and v1.4.3
No version of PrestaShop below v1.4.4 supported the automatic upgrade until v1.4.4. Therefore, automatic upgrade is not available to you unless you install the autoupgrade module. It is still an experimental module, so try it at your own risks!
You can download the module, and follow its instructions, on this forum thread. Please note that you need to be logged in the forum in order to download the module. Once installed, follow the instructions from this thread.