What is a forward/reverse proxy?

Diagram: a client in its internal network talks to a forward proxy, which connects over the internet to a proxy in front of the server in another internal network — Image created by Martin Thoma.

A proxy forwards requests. They can filter or change requests.

A forward proxy takes a request for the user and forwards it to a website. Examples are:

Privoxy: A forward proxy which helps to protect privacy (German Talk from 2016). You can also block advertisement with it.
Charles Proxy: Allows you to monitor and manipulate traffic. Charles Proxy can be used for testing.

A reverse proxy takes a request for a website and forwards it to a server. They can be used for caching and load balancing. Examples for reverse proxies are nginx or the Apache HTTP Server with mod_proxy.