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 while the Zotero stand-alone client is open, whatever you're looking at in your browser -- whether it's an article, book, webpage, etc -- will be saved into Zotero.
Any time you are on a web page which you'd like to save to Zotero, just click the Zotero Connector button 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 saves a copy, or "snapshot," of the web page as it appears at that moment. This copy is stored on your computer (and. optionally, in your Zotero.org online account). 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, in the Zotero client right click a saved web page reference and choose "View Snapshot":
In many popular databases, the Zotero Connector will not only save the details of the article you're looking at, it will even save the PDF of the article's full text! All you need to do is view the details of the article, then click the Zotero Connector button:
Back in the Zotero client, you can right click the item and choose "View PDF" to see the stored PDF copy: