What does a handshake mean when talking about computers?

In simple terms, it is the establishment of communication between computers. People can establish communication through a handshake, so the term is based on that. For computers to communicate they first must establish a connection with each other. It is like when you make a phone call and the other person answers, at which point communication has been established. One computer sends a message to the other and the other responds, opening a connection between them and that is what we call a handshake.