Standard old fashioned fairy lights are wired in series- that is the voltage has to go through each and every bulb to complete the circuit. So if any one bulb fails to light the whole thing does not work. You have to find the one that doesn't work and replace it. Sometimes the bulbs are OK, one has just come loose is all. Sometimes the first bulb nearest the plug is a fuse bulb, it is white coloured and different to the others. That is designed to fail first, you should have spares of this and should always check this FIRST.
Some of the new sets of lights that are in like sheets to cover windows have different wiring and are controlled by thyristers and things so they flash on and off and do fancy things. I could probably work these out but I am too lazy- when these go wrong I save some of the bulbs , bin the rest and buy a new set instead.
Next year I am going to buy sets of LED lights instead they are more reliable and use less power.
Here endeth the lesson