Check the bulb in your light fitting by replacing the bulb with one you know works before troubleshooting. It may simply be a blown bulb, rather a fault in the 3-way circuit. If it's the bulb then the problem is solved. If it's not the bulb, then complete the following steps.
Familiarize yourself with the principles of a 3-way circuit before starting to troubleshoot. A hot wire from your electrical panel connects to a terminal on one of the two switches. A second neutral wire goes directly to the light fixture from the electrical panel. Inside the first switch are two other terminals that have two wires connected. These are called travelers. The two travelers connect directly to two terminals on the second switch. A hot wire is connected to a third terminal in the second switch and goes to the light fixture, completing a circuit. The electricity flows through one or other of the two travelers, depending on the position of the switch.
Turn off the electricity for the 3-way circuit at the electrical panel. The panel has circuit breaker switches and each is labeled so you can identify the circuit that needs to be turned off.
Remove the covers from both switches in the 3-way circuit. There are usually a couple of small screws holding the cover in place, so use a screwdriver and remove the screws. Pull off the switch covers using your fingers.
First examine the inside of the switch nearest to the electrical panel The terminal that has a black wire and a black screw is the hot wire from the panel. The other two terminal screws are made of brass and usually have a black and a red wire attached; these are the traveller wires and go directly to the second light switch. However, you may find both traveller wires are the same color, such as both red or both black. So long as the two wires are connected to the brass terminal screws, they are the traveller wires.
Look at the inside of the second switch. The red and black wires that connect to the brass screws are the traveller wires from the first switch, but as indicated in the previous step, both wires may be the same color as each other. The black wire connected to the terminal with a black screw is the hot wire that goes to the light fixture. Once you're familiar with the wiring, turn on the electricity at the service panel.
Return to the previous switch nearest the electrical panel. Turn it on and set your multimeter to read the voltage. Move the dial and refine the voltage setting so it's set to 120 volts; your home electricity supply is between 110 and 120 volts.
Put the prong on the end of the black wire from the meter onto a metal part of the switch. The metal part of the switch is wired to ground in case of an electrical fault. Put the prong on the end of the red wire onto the black wire that's connected to the terminal with a black screw. The prong needs to touch the bare copper part of the wire.
Read the meter. A reading between 110 and 120 volts indicates power is getting to the first light switch. If there isn't a reading, there is a fault in the wiring between the panel and the switch. Contact an electrician to sort out the problem.
Place the red prong on one of the brass screws; it doesn't matter which one, but ensure you keep the black prong on the metal switch casing. Read the meter. You will get a reading if the switch is set so electricity is flowing through the switch from the hot or live wire to this traveller wire. If there isn't a reading, move the switch on the front of the switch cover to the other position so electricity flows to this traveller wire. If the meter doesn't register a reading regardless of the position the switch is set to, you need to replace the light switch as it's faulty. It there is a reading, repeat the process on the remaining brass screw. The meter should indicate a reading when the light switch is in one of the two positions. If there isn't a reading in either position, then replace the switch.
Repeat the entire procedure for the second light switch, but test the two traveller wires connected to the brass screws first. As before, you get a meter reading for both, depending on the position of the light switch. If you don't get a reading, then replace the switch. Test the black wire connected to the terminal with the black screw. You get a reading on the meter. If you don't then move the switch to the other position and you get a reading. If you still don't, replace the switch.
Replace the covers on the light switches, even if you intend to replace the switches. Use the screws to hold the covers in place.