The answer is rather simple: traditional Christmas lights (I'm ignoring newfangled LED varieties) were typically connected directly to the mains power supply and wired in series like this:
Only if the filaments of all the bulbs are intact will a current flow around the circuit; if one bulb breaks then the circuit is broken and all the lights go out. The reason the bulbs are wired in this, inconvenient, manner is that it's convenient for the manufacturer.
Although the supply voltage is 230v (or 110v) the bulbs are rated for a much lower voltage. At home I have a string of 20 lights like this with 12v bulbs. This works because of the rules of series circuits. In my home lights there are 20 bulbs each with some unknown resistance R. The total resistance of the circuit is 20R and the entire circuit is a sort of voltage divider.
The current flowing through the entire circuit is I = 230/20R and the voltage across any individual lamp is V= R * 230/20R or 230/20. So my 20 bulbs are each getting 11.5v. That's handy for the manufacturer because they can use cheap, small bulbs that use a low voltage.
BTW Some bulbs have a second piece of wire called a shunt that passes current when the filament breaks. With a shunt the manufacturer can still use series wiring and cheap bulbs, but a blown bulb doesn't stop all the lights from working.
Finding the broken bulb
A really fast way to find which bulb is broken is to perform a binary chop. To do that you need a multimeter (or similar meter to test continuity).
0. Unplug the string of lights from the power.
1. Remove the first and last bulbs and check that they are ok.
2. Remove the bulb in the centre of the string of lights. Using the multimeter check to see if there's an electrical connection between the contacts in the centre bulb socket and each of the end bulb sockets that you remove the bulbs from (you can actually look at the wiring to see which way the wires go and which contact that corresponds to).
3. Pick the half where there's no connection. The broken bulb is there. Remove the bulb that's in the middle of that half of the string and check it. If it's ok proceed to checking the electrical connection between the socket of the bulb you just removed the two nearest bulbs you removed (which will be the middle of the string and one end).
4. Proceed like that following where there's no electrical connection and dividing in half until you find the broken bulb.
This is a technique from computer science and you will find the broken bulb much faster (on average) than if you proceed checking each bulb in turn.