Term used to describe the process of one computer establishing a connection with another computer or device. The handshake is often the steps of verifying the connection, the speed, or the authorization of the computer trying to connect to it. An example of handshaking is when a modem connects to another Modem; the tones heard after the dialing is the handshake and is how the computers greeting each other.
If you did not grow up on a dial-up Modem or you're nostalgic, you can listen to a Modem connecting to the Internet in the below sound file. Each time someone wanted to connect to the Internet this is the noise their Modem would make.
In this audio file, you hear the Modem dialing a phone number and then communicating with the other Modem over the phone line. The squealing noise heard after the phone number is the Modem establishing a connection (handshaking). Once the connection is established the Modem goes silent.