What is a filter?
In short a filter is something stops you from viewing a certain website
An internet filter is software that is configured usually with a database list of URLs that are categorized. (For example When you visit a website such a Myspace it will be blocked as a "social networking/community forums/" type of website.) Sometimes instead they will use program that scans the content "live" as it comes threw the network or use a both live scanning and url database. These act as gateways or as a proxy (A real one), that intercepts the traffic before it reaches your computer. The entity or person running the network is the authority who decides which categories are to be blocked or not, but probably more so then not the person who bought the software uses the standard installation and does not customize the blocked categories and "over blocks" when their were just meaning to block porn, I've found that many of these software applainces do just that. There are numbers of such vendors will sell filter software. None of which I will mention here.
How they categorize websites?
With being running Proxy sites for some time now, I've researched and discovered most use categorization based on Bayesian examination of the site, Certain words have particular probabilities of occurring in websites. For instance, video sharing sites frequently have the word "Video".
Why do they use filters?
Filter vendors are simply filling a demand and promising "increased productivity" in the workplace and School places. As many people spend time using the internet for non-productive use, as such employers look for ways to increase productivity so they go to a filter vendor which is all to ready to sell them censorship software. For the same purpose, people are filing demand to circumvent their software.