There's already a question How exactly does a remote program like team viewer work which gives a basic description, but I'm interested in how the comms works once the client has registered with the server. If the client is behind a NAT then it won't have its own IP address so how can the server (or another client) send a message to it? Or does the client just keep polling the server to see if its got any requests?
Are there any open source equivalents of LogMeIn or TeamViewer?
The simplest and most reliable way (although not always the most efficient) is to have each client make an outgoing TCP connection to a well-known server somewhere and keep that connection open. As long as the TCP connection is open, data can pass over that TCP connection in either direction at any time. It appears that both LogMeIn and TeamViewer use this method, at least as a fall-back. The main drawbacks for this technique are that all data has to pass through a TeamViewer/LogMeIn company server (which can become a bottleneck), and that TCP doesn't handle dropped packets very well -- it will stall and wait for the dropped packets to be resent, rather than giving up on them and sending newer data instead.
The other technique that they can sometimes use (in order to get better performance) is UDP hole-punching. That technique relies on the fact that many firewalls will accept incoming UDP packets from remote hosts that the firewalled-host has recently sent an outgoing UDP packet to. Given that, the TeamViewer/LogMeIn company's server can tell both clients to send an outgoing packet to the IP address of the other client's firewall, and after that (hopefully) each firewall will accept UDP packets from the other client's Internet-facing IP address. This doesn't always work, though, since different firewalls work in different ways and may not include the aforementioned UDP-allowing logic.