The term half-open refers to TCP connections whose state is out of synchronization between the two communicating hosts, possibly due to a crash of one side.
According to RFC 793, a TCP connection is referred to as half-open when the host at one end of that TCP connection has crashed, or has otherwise removed the socket without notifying the other end.
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).
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.