TCP half-open
A half-open connection refers to a TCP connection that is partially open, which is in the process of being established.
The TCP protocol has a three state system for opening a connection. First, the originating site (A) sends a SYN packet to the destination (B). A is now half-open, and awaiting a response. B now updates its kernel information to indicate the incoming connection from A, and sends out a request to open a channel back (the SYN/ACK packet).
At this point, B is now "half-open" (it has sufficient information to receive packets, but not enough to send packets back). Note that B was put into this state by another machine, outside of B's control.
Under normal circumstances (see denial-of-service attack for deliberate failure cases), A will receive the SYN/ACK from B, update its tables (which now have enough information for A to both send and receive), and send a final ACK back to B.
Once B receives this final ACK, it also has sufficient information for two-way communication, and the connection is fully open.
See also
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Windows Xp Service Pack 2 Limits the number of half open connections you can have at one time to 10. By limiting the number of half open connections, you limit the rate at which new connections can be established. You don't limit the number of max connections.