An open proxy is a proxy server which any Internet user can use. Generally, a proxy server allows users within a network group to store and forward Internet services such as DNS or web pages, so that not as much bandwidth is used and it is more controlled. With an "open" proxy, however, any user on the Internet is able to use this forwarding service.
By using some open proxies ("anonymous" open proxies), users can hide their true IP address. Sometimes people will abuse open proxies, and so they are often seen as a problem. Anonymous open proxies will also make the user more anonymous and secure when browsing the web or using other internet services: a user's true IP address can be used by a hacker to get information from the user's computer.