Ohm's Law
Ohm's law is the fundamental rule for of all electric equations. It states that the product of current and resistance is voltage. For example, if a load with a 10-ohm resistance has 10 amps flow through it, then the voltage across the load is their product: 100 volts. If an ammeter were alone in series in with a large voltage, and it had its characteristically small resistance, then the current through it would also be large.
Current Divider Circuit
If two loads are in parallel with each other, then they begin at the same point and end at the same point. Current flowing into the loads will split, or divide, according to the relative strength of the resistances. Since current finds the easier path, more current will flow through the lower resistance than the larger resistance. On the other end of the parallel load, the two currents sum together and equal the original input current.
Shunt Amp Meter
So as not to overload the meter's internal circuitry with a very large current, all meters contain shunts. Inside the meter, there's always a resistor in parallel with the meter. This is the "shunt" resistor, and has a lower resistance than the meter. If a large current enters the meter, one that would damage it, the meter "shunts" the extra current through the shunt resistor.
Contrast with Voltmeter
Voltmeters, which measure voltage as electricity passes through them, also contain an extra resistor. In contrast with an meter, however, their "multiplier" resistors are in series with the meter and have high resistances. The high resistance creates a higher voltage across the shunt and a smaller voltage across the meter itself. This allows the meter to measure a total voltage much higher than what it can handle if it were alone in the circuit.