Syncing your library is helpful if you work on more than one device.
Let's look at a few examples:
Example #1:
You can install Zotero on your two computers, the one at home and the one at work, and setup synching in the application's settings on both computers, so no matter where you're collecting sources (by clicking the icon in the browser), your sources are being synced to the other computer because you are saving to the zotero.org server which updates your Zotero library/database on each computer.
Example #2:
You're doing research on a library computer, and you've added the Zotero connector to the browser and set it to save citations without using the desktop app. As you work, you save items by clicking the icon in the browser. Since you're not using the desktop app, all of the items you save are actually saving to the Zotero.org server. (you will need to be logged into your Zotero.org account and authorize the browser connector to access your Zotero.org account (see below)).
Later, at home, you open up the Zotero application on your personal computer. How will all of the citations you added at the library get to your computer? By syncing your library.
In the Zotero preferences, the default is to automatically sync your local library with the information on the zotero.org servers. This way, when you get home and open up the Zotero app it will automatically download the new items that you saved on the other computer at the library. (You'll know this is happening because the sync icon will be spinning.)
After installing the Zotero browser connector in your web browser, you'll need to connect it with your Zotero desktop application. This will allow the information you save while browsing to automatically appear in the Zotero app (the Zotero desktop application must be open) on your computer.
You can also choose to allow the browser to save items to Zotero.org if you need to work without access to the full Zotero application.
If you want to access the original GC Library electronic resource from within the Zotero application you will need to add the GC's proxy prefix to the Browser Connector.
From the Zotero Browser Connector preferences/options window click on the Proxies tab. Note: this window might appear different in different web browsers.
Click on the plus (+) button under the Configured Proxies
For Login URL Scheme enter: https://ezproxy.gc.cuny.edu/login?qurl=%u
For Proxied URL Scheme it is recommended that you use the %u option, so enter: %h.ezproxy.gc.cuny.edu/%u
When completed your settings should appear to similar to this: