This is the basic circuit for a three-way switch. The gray circle represents a light bulb controlled by the two switches. It is gray because it is "off." The two lines ending near the left side of the drawing go to a power source, like the circuit breaker panel in your house.
The green rectangles represent the switches. Notice that one conductor comes into each switch, but two go out. When the toggle is thrown the pathway inside the switch shifts from one of the out conductors to the other.
Here you can see that electricity can flow along the upper wire through the first switch, but its pathway is broken at the second switch and the light remains "off."
Step 1: The Light Comes "On"
Step 2: Turning the Light "Off" from another Switch
In my mind I like to think of three-way switches as like a construction zone on a four lane highway. If one set of lanes is closed, a crossover in the median moves traffic over to one of the lanes normally used for traffic going in the other direction. If there were no second crossover, traffic would stop. But, the second crossover brings traffic back to the original lanes and traffic continues to flow.