There are two types of upgrading Joomla:
The first one is making a minor update. This means, that you will upgrade only the build version of your application. For example, if you are upgrading your Joomla from one 3.4.x version to another.
To proceed with this update,
Step 1: You will need to download the correct patch and extract all the files on your local computer.
Step 2: After that, you will need to upload them on the server via FTP or from your cPanel -> Files -> File Manager. When uploading these files, please make sure that you will navigate to the directory where your Joomla application is installed.
The second one is a major migration. For example, this is an upgrade of your Joomla from 2.5 to 3.o version. For this type of migration,
Step 1: We suggest using the following Joomla component:
http://extensions.joomla.org/extensions/extension/migration-a-conversion/joomla-migration/jupgradepro
Step 2: After you have installed this component, you can proceed with the upgrade process from your Administration Area -> Components -> JUpgrade from where you will need to follow the provided instructions on the screen in order to upgrade your Joomla application.
Also, you can use a built-in function in new ever Joomla versions from where you can upgrade your Joomla automatically. You can use this function by accessing your administration area -> Components -> Joomla! Update.