A cursory inspection of a threeway switch will cause you to notice some differences between this unit and its dipole cousin:  1) the unit is wider; 2) it has three screws plus ground; 3) two screws at one end are brass, one screw is different (usually black). 4) the ON-OFF lever does not say "ON" or "OFF".
The action of flipping the switch connects  the black screw to one  brass screw  or to the other.  The switches are used in pairs.  The name "three way" comes from three possible settings: 1) both  levers "UP"; 2) both levers  "DOWN" ; 3) one lever "UP" the other "DOWN".  Three way switches are commonly used at the top and bottom of a stairway  so that the lights may be turned on or off from either  location. 
Each switch has one black screw and two brass screws  See figure below .  For each switch we should occupy the black screw first.  At one end, the black from power goes to the black screw; at the other end, the black from the component (in our case, a lamp) goes to the black screw.  Leave the corresponding white wires unattended for now.
The two switches are connected by a special cable known as the "traveler"  that has three conductors: black, red and white (plus ground).  At the switch near the power, connect black wire of the traveler cable to either brass screw, connect the red wire to the remaining brass screw,  connect traveler white to power white by wire nut .  Connect grounds.  Connect wires to screws at the component end in similar fashion.

See the diagram above to understand how this arrangement works. Find the small black box (screw) connected by one of two arrows to a brass screw. The lamp will be "ON" if both arrows are left or both arrows are right. The lamp is "OFF" if one switch is left, the other, right. Notice how the switch set up interrupts the black wire; the white wire acts as a return.