In an ideal world, all the add-ons you would want are officially vetted, approved, and can be found on the official website, Addons.mozilla.org .
If you are in this category, you can simply go to the Add-ons manager tab in Thunderbird, using the toolbar entry then and enter the add-on's name into the search box.
Here is a search for Add-ons that mention "Contacts":
Once you locate an add-on you like, click on Install, wait for it to be downloaded, then click on Restart Thunderbird if prompted.
You may have been given an URL for the add-on, or the add-on you want isn't on the "official" page. Searching for it, as shown above, doesn't locate the your add-on.
Visit the webpage in your browser, right-click the download link with your mouse, select "save link as", and save the XPI file to a location on your computer that you can find again. As the download is temporary it is suggested you use either your downloads folder or your desktop.
To install a downloaded XPI file into Thunderbird you have two options.
Right-click the link and select Save As or Save Link As as discussed earlier.
You will be invited to download and save the file. Most browsers have a default download location, often your Desktop. Make a note of where it downloads to, and what it's called. Some add-on files have silly names like “latest-234.xpi” which is not very helpful when you're trying to find it later on.
Open the Add-ons window in Thunderbird. There's a button to the left of the search window. Click this and from the subsequent menu select Install Add-on From File.
You will need to navigate in the file picker to where you saved the XPI file and select it to complete the install.
If you are using Windows, you can even skip the desktop bit. Open your browser and Thunderbird side by side, with Thunderbird's Add-ons tab displayed. Drag the link directly from the browser onto Thunderbird's Add-on tab. The main advantage of this method is that you don't need to actually save a copy of the add-on on your hard disk before it is installed.