Authentication is a difficult task - computers have no way of knowing that they are 'the computer that sits next to the printer on the third floor' or 'the computer that runs the sales for www.dotcom.com'. And those are the matters which are important to humans - humans don't care if the computer is '10.10.10.10', which is what the computers know.
However, if the computer can trust the human to tell it which computer address to look for - either in the numeric or the name form - the computers can then verify that each other is, in fact, the computer at that address. It's similar to using the post office - we want to know if 100 Somewhere Street is where our friend Sally is, but the post office just wants to know where to send the parcel.
The simplest form of authentication is to exchange secret information the first time the two computers communicate and check it on each subsequent connection. Most exchanges between computers take place over a long period of time, in computer terms, so they tend to do this in a small way for the duration of each connection - as if you were checking, each time you spoke in a phone call, that the person you were talking to was still that person. (Sally, is that you? Yeah. Good, now I was telling you about the kids .. is that still you?)
It may sound paranoid, but this sort of verification system can inhibit what is called a 'man in the middle' attack - where a third party tries to 'catch' the connection and insert their own information. Of course, this relies on the first communication not being intercepted.
Public key encryption (see below) is the other common means of authentication. It doesn't authenticate the sender, but it does authenticate the receiver - and if both parties exchange public keys, and verify by some independant means that the key they have is the key of the party they wish to send to, it authenticates both.