Jamroom, its modules and skins are updated through the Marketplace. You don't download or upload files. Your site needs to be connected to the Marketplace (see Your jamroom.net Account and the Marketplace).
By default Jamroom checks for updates during its daily maintenance and emails master admins when there are some. That's the Check For Updates setting in ACP > Modules > Core > Marketplace > Global Config.
The Marketplace sets up each updated module itself, so there's nothing else to run.
Before a big update, such as a new Jamroom version, back up your site's files and database. For extra safety, turn on Maintenance Mode while you update.
ACP > Modules > Core > Marketplace > Tools > Release Archive keeps the previous versions of updated modules and skins. Restore one from there if an update causes a problem. Install History lists what was installed and when.
ACP > Modules > Core > System Core > Tools > Integrity Check checks that every module and skin is set up correctly, and repairs what it can. It can also check the database and the language strings, and reset the caches.
You don't need it after a Marketplace update. Run it when:
Reset Caches (in the same Tools tab) is the first thing to try when a change you made doesn't show on the site.