Flash Cookies are another name for Macromedia Shared Objects that are stored in hidden folders on your computer. They operate much the same as HTML Cookies by saving certain information about sites that you visit, mainly your settings on those sites. They have become to being refered to as flash cookies.
Without them, everytime you visit a flash site (eg), a flash chat service such as 2100chat.net, you would have to re-select your settings. I hope I explained this to satisfy your question.