If a single bulb on one of the two parallel circuits has gone bad, it will shut down the one circuit, causing half the set to go dark. That is, if the lights are wired with two parallel circuits, which is the most common type sold.
Most strings of Christmas lights are wired to run on a circuit. Electrical current runs through each bulb on the string, in order to complete the circuit. If even one bulb is faulty, the circuit is broken and the lights will not operate.
If only half of the string will light up, that probably means the set contains two different circuits, each circuit operating one half of the string. One circuit has remained intact, but the other circuit is broken somewhere along the string.
The culprit may be a loose bulb. The string should be checked for any bulbs that are not screwed in tightly. If there are no loose or obviously damaged bulbs, the problem may be a broken filament. Each bulb on the dark half of the string will have to be removed and tested with a voltage tester, to make sure it is capable of passing current. Once the bad bulb is located it can be replaced with a new one.
Another possible culprit is a bad shunt in one of the bulbs. Some string lights contain shunts, which are designed to activate when a filament burns out, and pass the current along the circuit. When a shunt fails to activate, it breaks the circuit.