This is a proxy server in a nutshell. It's a client app (it could be a web browser, a website that acts as a web browser, and so forth) as well as a server that's client requested. The proxy server is a program that provides networking capabilities for users to make indirect network links to other network services, hence the term "proxy"; instead of you directly accessing something, it serves as your proxy. The proxy accesses whatever site (usually something that's normally inaccessible to you) for you in cache while also masking your IP address since its own IP address is being used to gain network access.
This server then captures and diverts all client requests to your actual server to gauge if it can satisfy these requests for itself. If that's not the case, then it sends the request to the real server instead. There are also cases wherein the server's response or the client's request may be altered for the proxy depending on the circumstances for a variety of reasons. Proxy servers are mainly used for two specific goals. The first one involves performance improvement. Your computer's performance can be improved with the use of a proxy because it caches and stores accessed information for an amount of time, which means repeated access is much faster with it.
Every time you request to visit the same forum, site, bulletin board, image board, content aggregator, blog, and whatnot, it will retrieve the saved pages except during those times wherein the content is changed or dynamic (and during such situations, the proxy will resolve things by double-checking and accessing the site again for any new content that isn't part of its saved cache). The second main purpose of using a proxy server is request filtration. These servers are capable of filtering requests, and its practical applications include preventing employees from accessing certain websites, or banning them outright in case they serve to lower productivity (like a Facebook or YouTube ban).
In light of these main purposes of (presumably propriety) proxy servers, it's therefore ironic how free web-based anonymous proxies are used by users in order to bypass request filtration (i.e., in order to bypass a proxy, use another proxy with a separate IP address that isn't keeping you from accessing a given forum or blog). More to the point, to the layman, the most common use of free web proxies is to overcome (instead of instigate) website filters, IP bans, or region-bans on certain videos (for example, some YouTube videos are not available for viewing in Germany or the United States thanks to copyright holders withholding access to them).