When you click on the Quote button on another users post, you get the entire post you are replying to in between BOLD tags [ B ]. In front of that you get a short blurb about the post you're replying to between ITALIC tags [ I ]. All this is included between QUOTE tags [ Q U O T E ].
To remove parts of the original post that you don't want to quote, simply highlight what you want to remove by dragging with your mouse, and then hit the Delete key. Make sure you don't highlight and remove part of the BOLD tag on either end.
You can also insert the start and stop QUOTE tags yourself, before and after something you type in or copy and paste in, like:
Four score and seven years ago...
You can do the above by using the Quote button and typing in the text in the little box, or you can just type [ Q U O T E ] (without the spaces) at the start of the text to be quoted and [ / Q U O T E ] (without the spaces) at the end of it.
Hope this helps,