The first thing you need to do is make a backup of your website files and database. Then you will need to find what version of e107 you are currently running. This can be found in the 'Site Info' panel in your e107 admin area homepage.
You can also (If not set already) enable version update checking in the 'Preferences' > 'Advanced Features' section. This will notify you when a new version of e107 has been released.
OK, so you need to update... here's how to do it correctly.
Check your current e107 version and go to the e107 Downloads page on the developers website. You now need to make sure you download the correct files. At the top should be the latest stable version, you do not need to download the full version files. Overwriting your site files with the full version will break your website.
If you are running a version of e107 older than the previous release, eg: v0.7.15 and latest is v0.7.18 then you need to download the files that cover culmative updates so, v0.7.x to latest version upgrade. (v0.7.x to v0.7.18 for example)
If your current version of e107 is the one before the latest version, eg: v0.7.17 upgrading to v0.7.18, then you just need the v0.7.17 to v0.7.18 upgrade files.
Once you have downloaded the .zip file containg the correct upgrade, create an empty folder on your desktop and extract the files to this folder.
Connect to your ftp using the recommended method outlined in the other tutorial and upload the files to your 'public_html' directory.
