Repeaters expand the point-to-point range of radio communications.  Most repeaters you will use in the radio world have a few things in common.
|•||High elevation by being atop a tall building, hilltop or on a radio tower.  By placing the repeater at a high elevation, the RF horizon is expanded.  It also helps reduce the amount of interference caused by buildings, forests and topology.|
|•||By having a stationary repeater, you have access to power mains and often battery backup systems that would not be practical in a handheld or even a mobile unit.  This means the transmitter can put out optimal power to cover horizon to horizon with enough signal that the less sophisticated handheld receivers can detect.|
|•||Being stationary means size and weight restrictions don’t apply the same way they would in a portable or handheld system so the receiver circuitry can be more complex and sensitive to the weaker signals put out by handheld units.|
Point-to-point (simplex) operation is preferred if you are within range of the other station.  In these examples we are using the 2 meter and 70 cm bands, but the same principals apply to other bands as well.  In this case, unit A and Unit B can communicate in both directions with a direct simplex connection (both using the same frequency).
In the diagram above, the traffic from unit A is shown in red, while the return traffic to A is shown in green.  Unit A might have trouble reaching unit C due to the forest, unit B due to the mountainous terrain and the mobile (vehicle mounted) unit because of the buildings.
The relatively weak signal from unit A reaches the repeater R on one frequency.  The repeater retransmits or passes the signal through to a different frequency usually at a higher power level.  The reason the frequencies are different is that if it retransmits on the same frequency, the signals would interfere with each other.  There is a minimum separation for the send and receive frequencies in the repeater because otherwise the transmitted signal would still interfere with the received signal.  There are some specific conventions used in radio to determine the separation as well as what frequencies are used for each side of duplex repeater communications.  When a repeater is set up, the frequency coordinator for the region (a volunteer) helps to choose the frequencies in order to prevent repeaters from interfering with each other.
In some circumstances, you may find you are unable to reach the main repeater for the area from your handheld device.  Perhaps you don’t want to use the high power setting because you don’t want the battery to run down too quickly.  This is where having a dual-band mobile radio in your vehicle capable of cross-band repeat comes in handy.  This allows you to take advantage of the much larger battery and almost certainly better antenna in your vehicle to reach the repeater.
The term cross-band refers to the fact that your mobile unit is receiving on one band and retransmitting that signal on the other band.  The less sophisticated filters in mobile radios make retransmission on the same band as the receiving band problematic without costly external antenna duplexers and multiple antennas.  It is referred to as an auxiliary repeater because it is not coordinated with the frequency coordinator in your region.  There is space set aside on the VHF and UHF bands for auxiliary repeaters.
In simplex or 1-way cross-band repeating, your local frequency is only used to talk to your mobile radio, while your receive frequency is tuned directly to the main repeater for the area.  FCC station identification rules apply and are met by the fact that your mobile radio is only transmitting on the uplink side to the area repeater and when you station ID from your handheld, that passes through your mobile unit so both have transmitted the station ID as required.
If you are having trouble receiving useable signal on your handheld from the area repeater, then you will need to configure your mobile unit for duplex or 2-way cross-band repeat.
This allows the use of a “stubby” antenna on your handheld in situations or activities where even a short whip antenna would be problematic.  In this scenario, your handheld is programmed to communicate with your mobile radio simplex.  Your mobile radio retransmits what you transmit to the local repeater, and retransmits what it receives from the local repeater to your handheld.  Generally speaking, an auxiliary repeater will assign whichever side breaks squelch first as the master until the carrier drops and it goes back to seeing which side goes active first once again.
A large disadvantage of the 2-way auxiliary repeater is Station ID requirements.  As with the 1-way repeater mode, the mobile radio in this example would satisfy the FCC requirements for station ID on the main repeater channel when you give the station ID of your handheld.  There is not a mechanism in many radios capable of cross-band repeat mode that will station ID in both directions.  The FCC requires that you identify your station upon every frequency currently being transmitted on.  Fortunately, most handheld dual-band radios have two transceiver units or variable frequency oscillators (VFO’s) and your b-channel side can be set to the same transmit frequency as your main area repeater’s transmit frequency with the power at the lowest setting.  When it is time to perform a station ID you can “back feed” the station ID through your mobile unit in this way, then switch your mobile back to the primary channel.