Types

Intercepting proxy server
An intercepting proxy (also known as a "transparent proxy") combines a proxy server with a gateway. Connections made by client browsers through the gateway are redirected through the proxy without client-side configuration (or often knowledge).
Intercepting proxies are commonly used in businesses to prevent avoidance of acceptable use policy, and to ease administrative burden, since no client browser configuration is required.
It is often possible to detect the use of an intercepting proxy server by comparing the external IP address to the address seen by an external web server, or by examining the HTTP headers on the server side.

[edit] Transparent and non-transparent proxy server
The term "transparent proxy" is most often used incorrectly to mean "intercepting proxy" (because the client does not need to configure a proxy and cannot directly detect that its requests are being proxied).
However, RFC 2616 (Hypertext Transfer Protocol -- HTTP/1.1) offers different definitions:
"A 'transparent proxy' is a proxy that does not modify the request or response beyond what is required for proxy authentication and identification".
"A 'non-transparent proxy' is a proxy that modifies the request or response in order to provide some added service to the user agent, such as group annotation services, media type transformation, protocol reduction, or anonymity filtering".

[edit] Forced proxy
The term "forced proxy" is ambiguous. It means both "intercepting proxy" (because it filters all traffic on the only available gateway to the Internet) and its exact opposite, "non-intercepting proxy" (because the user is forced to configure a proxy in order to access the Internet).
Forced proxy operation is sometimes necessary due to issues with the interception of TCP connections and HTTP. For instance interception of HTTP requests can affect the usability of a proxy cache, and can greatly affect certain authentication mechanisms. This is primarily because the client thinks it is talking to a server, and so request headers required by a proxy are unable to be distinguished from headers that may be required by an upstream server (esp authorization headers). Also the HTTP specification prohibits caching of responses where the request contained an authorization header.

[edit] Open proxy server
Main article: open proxy
Because proxies might be used for abuse, system administrators have developed a number of ways to refuse service to open proxies. Many IRC networks automatically test client systems for known types of open proxy. Likewise, an email server may be configured to automatically test e-mail senders for open proxies.
Groups of IRC and electronic mail operators run DNSBLs publishing lists of the IP addresses of known open proxies, such as AHBL, CBL, NJABL, and SORBS.
The ethics of automatically testing clients for open proxies are controversial. Some experts, such as Vernon Schryver, consider such testing to be equivalent to an attacker portscanning the client host. [1] Others consider the client to have solicited the scan by connecting to a server whose terms of service include testing.

[edit] Reverse proxy server
Main article: reverse proxy
A reverse proxy is a proxy server that is installed in the neighborhood of one or more web servers. All traffic coming from the Internet and with a destination of one of the web servers goes through the proxy server. There are several reasons for installing reverse proxy servers:
Encryption / SSL acceleration: when secure web sites are created, the SSL encryption is often not done by the web server itself, but by a reverse proxy that is equipped with SSL acceleration hardware. See Secure Sockets Layer.
Load balancing: the reverse proxy can distribute the load to several web servers, each web server serving its own application area. In such a case, the reverse proxy may need to rewrite the URLs in each web page (translation from externally known URLs to the internal locations).
Serve/cache static content: A reverse proxy can offload the web servers by caching static content like pictures and other static graphical content.
Compression: the proxy server can optimize and compress the content to speed up the load time.
Spoon feeding: reduces resource usage caused by slow clients on the web servers by caching the content the web server sent and slowly "spoon feeds" it to the client. This especially benefits dynamically generated pages.
Security: the proxy server is an additional layer of defense and can protect against some OS and WebServer specific attacks. However, it does not provide any protection to attacks against the web application or service itself, which is generally considered the larger threat.
Extranet Publishing: a reverse proxy server facing the Internet can be used to communicate to a firewalled server internal to an organization, providing extranet access to some functions while keeping the servers behind the firewalls. If used in this way, security measures should be considered to protect the rest of your infrastructure in case this server is compromised, as it's web application is exposed to attack from the Internet.