You do not need to create an account to install or use Zotero. However, creating an account allows you to view your "web library" online when you are away from your main computer and sync your library between two or more computers. The web library can also be used to save resources with the Zotero connector on computers that don't have the desktop application installed (as with Chromebooks or computers you can't install software on). Register for a free account.
Zotero cloud storage is free up to 300MB, which will be enough space for almost all undergraduate work and even most graduate work. If you are reaching the limit, you can pay to increase your storage. However, you should also clean up your collections to remove any duplicates that may be unnecessarily increasing your storage. Learn more about storage limits and pricing.
Note: Creating an account will not automatically store your PDF attachments online or sync PDF attachments between computers. If you are syncing between multiple computers, it is a good idea to run the sync before you exit Zotero on each computer and again first thing when you open Zotero on each computer. Otherwise, you may not have matching libraries.
Zotero recommends using a backup utility to save backups of your entire computer. This is absolutely the best practice to ensure that you don't lose any of your work. It is even better to save a backup to an external hard drive - even just doing this weekly may save you a lot of work if your computer completely crashes. You can also upload a weekly backup to a cloud service.
However, you may want to save a backup of just your Zotero library when you are performing a new install. This may be necessary if you are using a new computer and don't want to use the Web Library or if you are trying to troubleshoot an issue and don't want to lose your data. The Zotero desktop standalone application saves all of your library in a couple of folders on your computer. You can save all of the work under one folder - the Zotero folder - by making a copy of it and saving it to a new location. This is a two-step process of locating the Zotero folder and then saving a copy of that folder.