A circuit-level gateway blocks all incoming traffic to any host but itself. Internally, the client machines run software to allow them to establish a connection with the circuit-level gateway machine. To the outside world it appears that all communication from your internal network is actually originating from the circuit-level gateway.
A proxy server is generally put in place to boost performance of the network, but can act as a sort of firewall as well. Proxy servers also hide your internal addresses as well so that all communications appear to originate from the proxy server itself. A proxy server will cache pages that have been requested. If User A goes to Yahoo.com the proxy server actually sends the request to Yahoo.com and retrieves the web page. If User B then connects to Yahoo.com the proxy server just sends the information it already retrieved for User A so it is returned much faster than having to get it from Yahoo.com again. You can configure a proxy server to block access to certain web sites and filter certain port traffic to protect your internal network.
An application gateway is essentially another sort of proxy server. The internal client first establishes a connection with the application gateway. The application gateway determines if the connection should be allowed or not and then establishes a connection with the destination computer. All communications go through two connections- client to application gateway and application gateway to destination. The application gateway monitors all traffic against its rules before deciding whether or not to forward it. As with the other proxy server types, the application gateway is the only address seen by the outside world so the internal network is protected.