A link light is a handy little LED
typically found on network devices. The purpose of it is to show if the device is linked to another device on the other end of the cable. In other words, if the link light is on, then there is a connection, and the physical network is fine, but if there is no link light, then you know there is probably a problem along the line (eg no power, faulty cable).
Obviously, this has substantial uses in network troubleshooting, and is one of the first things to have a look at if you seem to be having difficulty in accessing a network. If it is on, then the problem is likely a software one, if the link light is off, then you should look for a hardware problem. It should be noted there are a small number of exceptions to this, for example some network cards won't create proper link pulses unless their software drivers are properly loaded. However, it is still quite useful as a general rule.