The "quote" button which is contained within every post at the bottom right, allows a person to quote a part of a that post, for example; as if your reading a post and you want to respond to a specific point (sentence) made within that post, then that post you are reading, you would click "quote" to do so. You will then be opened up into the editor, with the existing post contained betweeen "quote" tags. You then remove the non-essential parts of the post, keeping only the point you want to respond, and you leave that point enclosed within the quote tags that you will see, being an opening tag, and closing tag. The closing tag contains the / signalling to the system to close the quotation.
By default when using the quote button, the system will automatically insert the username of that person you are quoting, and the post will be linked that it came from. If a quote started with [quote], then no name or link will showup. If a quote contains [quote=anthony], then the quote will show my username as the originating author, if the quote shows [quote=anthony;number here] then the quote will show me as the original author, plus a link to that post itself.
The system also incorporates a multi-quote, which besides the quote button you will see another botton that contains a quotation mark. If you where reading a thread and wanted to quote several posts in one, then you simply click those upon each post, when clicked each will highlight to a new colour showing the post has been selected, up to a maximum of five posts within any thread, you then click "post reply" which all posts you checked for multi-quoting will be presented between their respective quote tags. You would edit out the content of each leaving only the statement you are directly referring, responding to each quote, ensuring each one is left wrapped in its opening and closing quote tags.
You can read most of these type things in the FAQ of the forum, BBcode and other aspects contained in the forum.
The full manual to the forum is contained at:
http://www.vbulletin.com/docs/html/