Extensions are updated more frequently than the core Joomla! files and they can represent a large source of trouble. It is your job to keep up with the changes, not the developers. Even if they make a heroic effort to reach you, it's still your job.
Obtain the patches from the developer. (I suggest the service www.salvusalerting.com to stay on top of the patch and vulnerability information.
Make a full backup of your database and website and pull a copy off to another computer or a USB drive.
Review the particular extension's directions - but generally most extensions require removal of the old. You can try to do an install and *SOME* extensions will update right over the old version. You can review both methods below.
Make a full file and database backup.
Click Extensions | Install / Uninstall.
Install the extension.
Test site and extension.
Make another full file and database backup.
If the extension has been written to allow upgrading, then it will overwrite and not...