Feeds from household service panels and many commercial lighting panels are either two-wire or three-wire. The terms two-wire and three-wire refer to the number of insulated conductors in a cable, typically non-metalic sheathed cable in residential wiring. The two-wire feed is simple, one black hot wire, one white neutral wire and one bare bonding wire. The bonding wire connects to a bonding lug in the panel. The neutral connects to a neutral bar, which is grounded at the main point of supply, and the hot wire goes to an appropriately sized (ampacity not physical size) breaker in the panel.
Three-wire feeds add one more wire and potentially a whole lot more complexity. The extra wire in a three-wire feed is red. That red wire goes to a breaker in the panel, similar to the black wire. The key, too often missed by do-it-yourself-ers and even some electricians, is that the red wire must be on a different phase than the black wire.
A residential electrical service, while called single phase, has in effect two phases (though both are opposite halves of the same sine wave). From either phase to ground is 120 volts. From one phase to the other is 240 volts (may be 208 volts in multi-family or commercial installations). These are nominal values. Lots of people call them 110 and 220 volts. For a nominal 120/240 volt, single phase service BC Hydro considers a range from 110/220 to 125/250 volts acceptable.
In a two wire feed, current flows through the hot wire, through whatever loads are on the circuit, completing the circuit through the neutral. The current in the hot wire is equal to the current in the neutral wire. Assuming the circuit is not overloaded, everything is fine.
Three-wire feeds are used to feed loads like your range or dryer, which have both 240 volt (heating elements and dryer motor) and 120 volt (lights, clocks and what have you) loads. If the red wire and the black wire are not on different phases, your appliance will only receive 120 volts. Heating elements will only produce one quarter (yes, one quarter) of the power they should and your dryer motor may burn out from drawing too much current (yes, too much current at the lower voltage).
Three-wire feeds are also used to feed two 120 volt circuits from the same feed. When this is done properly, the neutral wire only carries current equal to the imbalance between the two circuits. Say the black wire had 9 amps going through it and the red wire had 11 amps, the neutral would carry 2 amps, the difference between the current in the black and the current in the red. If the three-wire feed is connected incorrectly, with both red and black on the same phase, the neutral would carry the sum of the two currents rather than the difference. From our example above, if the black wire had 9 amps and the red wire had 11 amps, in the incorrectly connected feed the neutral would carry 20 amps. The 14 guage wire normally used in residential wiring is rated for only 12 amps. In time an overloaded neutral such as in this example could result in a fire.
Beware of breakers that have two handles, but are only the width of one full-sized breaker space in the panel. These breakers may feed both circuits from the same phase. To use these breakers with three-wire feeds, the red and black wires of each three-wire feed must be connected to two separate breakers. Typically they will be connected to adjacent halves of two breakers. Depending on how those circuits are used, a tie-bar may be required across the two breaker handles. If the breakers supplying a three-wire feed are not adjacent, a special tie-bar may be used that spans the breaker handles between.
Another concern with three-wire feeds is what can happen if the neutral ever becomes open. This could happen because of someone working on wiring without turning off the circuits, or because of a poor splice or a wire being exposed to damage due to poor installation. The two circuits are each 120 volts to ground and to neutral (which is grounded at the service), and are connected to different phases, so have 240 volts between them. Loads on those circuits are connected between one phase and neutral, so receive 120 volts. If the neutral becomes open, under some circumstances the 240 volts available between the two circuits can be distributed depending on the relative impedances of the two circuits. It is even possible for one of the circuits to receive practically the full 240 volts while the other receives almost 0!