The question Amazon answers with it
"Can I Check If My Links Are Tagging to My Account?" is the heading of the Amazon Associates help page that explains this parameter. Amazon's answer: click your link and look for your Store ID or Tracking ID in the URL, which Amazon says will most often appear as &tag= followed by that ID. The tag is how a purchase gets credited to the Associate who sent the shopper.
Amazon's operating agreement calls these tagged links Special Links, and its policies say they must use the Associate ID Amazon assigned. The same agreement requires Associates to state "As an Amazon Associate I earn from qualifying purchases" where they use them. One caution: tag is a common word, and a site that is not Amazon can use the same key for something unrelated. This page covers Amazon's.
How long a tag counts
Amazon's Commission Income Statement defines the window. A session starts when the shopper clicks the tagged link and ends at whichever comes first: 24 hours pass, the shopper orders a product that is not a digital product, or the shopper clicks a tagged link that belongs to a different Associate. A product added to the cart during that session still earns commission if the order is completed within 89 days of the first click.
Checking a link you built
Links created in Associates Central are automatically coded to the Tracking ID you picked, and Amazon's Link Checker does not work with them. The Link Checker is for links you created or edited yourself outside Associates Central. Amazon's customer support will also confirm a site's links are tagged to you if you send the page URL.
If you are the one shopping
The product page itself does not change with or without the tag. Removing it changes who, if anyone, earns commission on what you buy, so keep it if you want the person who recommended the product to be paid. Other affiliate networks use their own parameters, such as awc for Awin and irclickid for Impact, and the link decoder labels each one it recognizes.