Installing an extension in Magento 1 or Magento 2 is easy and if done right can be safe to your store. If something does go wrong, take a look at the Uninstalling a Magento extension guide.
Installing a Magento 1 Extension
The easiest way to install a Magento 1 extension is using SFTP or FTP. To do this, first download your Magento module. This will usually be either a ZIP file or a TGZ file and extract the files on your computer.
Login to the Magento Admin and disable all caches (System > Cache Management). If you're using the Magento compiler (although why would you be?!) you need to disable it (System > Tools > Compilation).
Connect to your site using FTP or SFTP and merge the extracted module files with your Magento root. Once all of the files have been uploaded, go to any page on your website and the module will be installed.
To complete the installation, logout of your Magento Admin and log back in. You can now flush and then enable the Magento cache. If you want to re-enable the Magento compiler (I wouldn't recommend it, it won't make your site faster) ensure that you recompile before enabling.
Trouble Shooting
If the installation has gone wrong and you have a blank white page or the content stops half way through, check the Magento fatal error guide that explains how to reveal the error message.
Installing a Magento 2 Extension
The easiest way to install a Magento 2 extension is using SFTP or FTP. To do this, first download your Magento 2 module and extract the files.
Upload the files to your Magento website. Where you should upload the files depend on the module and how the module's archive was built. Module's generally live in app/code/VENDOR/MODULE but check the module's installation guide for the specific installation location.
Access the terminal and run the following command, replacing {{MODULE}} for the actual module name.
php bin/magento module:enable {{MODULE}}
php bin/magento setup:upgrade