Twisted-pair cabling uses one or more pairs of wires that are twisted together and enclosed in an outer durable jacket
When two wires are parallel and in close proximity, the electrical signal from one wire induces a similar electric signal in the other from the expanding and collapsing magnetic field associated with the electric field. This interference is called crosstalk. You can sometimes hear someone else dialing a phone or talking faintly when when you're calling Mum on the telephone... this is an example of crosstalk.
One way to reduce crosstalk is to move the wires further apart, which is not practical. The other way to reduce crosstalk is to twist the two wires together. The signal induced by one wire will be cancelled out by the other wire. This also helps to reduce the effects of outside interference.
As a practical example, the US Navy had a problem with aircraft wiring inducing unusual signals into critical flight-control systems. After one fatal helicopter crash near a radio tower, they researched the causes and found the radio tower had induced interference in the helicopter's flight-control system. Now, all military and commercial aircraft use either twisted pairs or coaxial cables for most of its wiring.
Another practical application is the cable used in Ethernet. These cables run very close to power lines, flourescent lighting fixtures, monitors and other "noisy" devices. By twisting the wires in the Ethernet cable, you can reduce crosstalk and external interference.
The two wires that are twisted together must be related, normally a signal line and its associated signal return path. Putting a 400Hz power line next to a voice line would induce a buzzing noise so loud that it would be unusable.