Reverse HTTP is a way for the client to keep an open connection to a web server so that the web server can push updates to the client (rather than the client continually having to ask for updates).
Take your classic Twitter client for example.
Currently, the client asks Twitter periodically if you have any updates. If not, it's a wasted request.
With technology like Reverse HTTP, once you establish the connection to Twitter...Twitter would be able to send you the updates when they happen saving both you and Twitter some bandwidth, overhead, and a little effort.
Reverse HTTP works by running a Web Server inside the browser that the server communicates with.
There are similar technologies that accomplish the same goal in a safer, more secure way. Microsoft.NET implements these kinds of services as Duplex Binding services in WCF by keeping the connection between the client and server open once it is made (instead of running a seperate server on the client). There's also a technology called Comet which allows the same thing.