After installing the Zotero Connector browser plugin (https://www.zotero.org/downloads) you will notice a new icon in your browser's toolbar. It will look like the letter Z, a piece of paper, a book, or so on, depending on the type of site you are on. If you click this, whatever you're looking at in your browser -- whether it's an article, book, webpage, etc -- will be saved as a reference in Zotero.
Note: Zotero Connector requires that the Zotero desktop application be open, or that you have Connector set up to sync to your Zotero web library. See the "Installing Zotero Connector" tab for more details.
Any time you are on a web page which you'd like to save to Zotero, just click the Zotero Connector icon in your browser's tool bar:
Back in the Zotero client you will see that this web page has been added to your list of items.
One more thing: when you click the Zotero Connector button, it doesn't just save the reference details of the webpage. It also attaches to the reference a copy, or "snapshot," of the web page as it appears at that moment. That means that even if the website you referenced is updated or taken offline, you will always have the copy you were looking at when you did your research!
To view a snapshot...
Using the Zotero desktop application: right click a saved web page reference and choose "View Snapshot"
Using the Zotero web application: click on a saved web page reference. In the right column, where the details of the reference are displayed, click "attachments." Then click the "Open" icon to the right of the word "Snapshot" (see screenshot):
In many article databases, Zotero Connector will not only save the details of the article you're looking at, it will even save a PDF of the article! All you need to do is view the details of the article, then click the Zotero Connector button:
To view a PDF attached to a reference in Zotero...
In the Zotero web app: click on a reference, then in the right-most column click "attachments" to see attached PDFs.
In the Zotero desktop app: you can right click the item and choose "View PDF"