When you click on Quote it will quote whatever post you clicked it under.
To use the multi quote tab click it under all the post that you would like to quote, and on the last post in addition to clicking multi quote click quote, and it will quote all the post that you choose.
Both Quote, and Multi Quote, quote the post on a different page. The Quote page give you all the options available when replying to a post including smilies and the option to add attachments like pictures.
When Quick Reply is clicked it takes you down to the quick reply box at the bottom of the page but does not quote the post it's clicked under. It's basically a way to skip down to the quick reply box without scrolling down to it. It also has limited options when replying rather then the full set of them that Quoting, starting a new thread, or clicking Go Advanced under the Quick Reply box will get you.
Clicking then thanks icon just puts that you told the person Thank You. You'll see it on the left side of whatever post you clicked it on.
I hope that helps a bit and I wasn't too confusing!