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The Firewalls and Internet Security seminar defines three basic types of firewalls: packet filters, circuit level gateways, and application gateways. Of course there are also hybrid firewalls, which can be combinations of all three.
Packet filter gateways are usually comprised of a series of simple checks based on the source and destination IP address and ports. They are very simple to the user since it will probably not even realize that the checks are taking place (unless of course it was denied!!). However, that simplicity is also their biggest problem: there is no way for the filter to securely distinguish one user from another. Packet filters are frequently located on routers and most major router vendors supply packet filters as part of the default distribution. You may have heard of smart packet filters.

Smart packet filters are really not very different from simple packet filters except they have the ability to interpret the data stream and understand that other connections, which would normally be denied, should be allowed (e.g. ftp's PORT command would be understood and the reverse connection allowed). Smart packet filters, however, still cannot securely distinguish one user on a machine from another. Brimstone incorporates a very smart and configurable application layer filter.

Circuit-level gateways are much like packet filters except that they operate at a different level of the OSI protocol stack. Unlike most packet filters, connections passing through a circuit-level gateway appear to the remote machine as if they originated from the firewall. This is very useful to hide information about protected networks. Socks is a popular de-facto standard for automatic circuit-level gateways. Brimstone supports both Socks and a manual circuit-level gateway.

Application gateways represent a totally different concept for firewalls. Instead of a list of simple rules which control which packets or sessions should be allowed through, a program accepts the connection, typically performs strong authentication on the user which often requires one-time passwords, and then often prompts the user for information on what host to connect to. This is, in some senses, more limited than packet-filters and circuit-level gateways since you must have a gateway program for each applications (e.g. telnet, ftp, X11, etc). However, for most environments it provides much higher security because unlike the other types of gateways, it can perform strong user authentication to ensure that the person on the other end of the IP connection is really who they say they are. Additionally, once you know who you are talking to, you can perform other types of access checks on a per-user basis such as what times they can connect, what hosts they can connect to, what services they can use, etc. Many people only consider application gateways to be true firewall, because of the lack of user authentication in the other two types. The core Brimstone ACL provides application gateway functionality.

Hybrid gateways are ones where the above types are combined. Quite frequently one finds an application gateway combined with a circuit-level gateways or packet filters, since it can allow internal hosts unencumbered access to unsecured networks while forcing strong security on connects from unsecured networks into the secured internal networks. Recommended Brimstone configurations are a hybrid firewall.

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