Firewalls come in several categories and sub-categories. The basic goal is to prevent intrusion, the difference is in how they try to accomplish this. The major categories of firewalls are Network layer firewalls and Application layer firewalls. The difference between the two is that the former operate at the low level of the TCP/IP? protocol stack as packet filters, not allowing packets to pass the firewall unless they meet the rules defined by the firewall administrator - while the later work as proxies on the application level - and may inspect the contents of packets, sanitize them, and so forth.
Network layer Firewalls generally fall into two sub-categories, stateful and non-stateful. Stateful firewalls also hold some information on the state of connections (i.e. established or not) as part of their rules (e.g. only hosts inside the firewall can establish connections on a certain port).
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