A quote, like it says, is some text that was written by someone else and that you would like to display in your own writing. Quoted text is displayed in a special format, as you can see on the image below.

Quotes in a message

This is commonly used in forums. For example, in a topic that contains a couple of posts, it often happens that you would like to comment or reply to something that a user wrote previously before. To do that, you would simply quote what the other user wrote, and then comment or reply to it.

The reason for quote tag is that it appears in a different format in the forums so readers clearly know that this is a quote.