Current only "flows" when there is a voltage to push it.
Actually, it's the other way around. Voltage exists when a current flows through a resistance. Many power supplies and amplifiers are voltage regulators which adjust the output current such that the voltage between the output terminals equals a set voltage.
And then the amount of current flowing is dependent on the impedance of the loudspeaker. Typically a 4 ohm loudspeaker will draw twice as much current as an 8 ohm loudspeaker for the same given amplifier voltage.
Some amps are capable of even more current into lower impedances. Whereas some equipment such as receivers cannot give up more current into 4 ohms and are only safely rated for 8 ohms.
Yes, the impedance of the loudspeaker comes into play in terms of average current, but where high current matters is when the desired output voltage is changed. In a square wave, the edge transition is extremely sharp and is basically an infinite slope. If you fed that to an amplifier, the amplifier will take some time to completely make the change and thus the output will be a square wave with a rounded leading edge. How fast the transition occurs is called the slew rate of the amplifier which is expressed as a rate of voltage over time.
Slew rate is governed by many factors, one of which is how much current the amp can provide to the loudspeaker. Remember that a voltage regulated amp works by controlling the output current such that the voltage across the loudspeaker meets a set point. When the set point is changed, the amplifier will quickly open up the output current to try to match the new set point. The drivers and crossover components within the loudspeaker provide inductance and capacitance that will need to charge sufficiently before the output voltage will change. If the amplifier merely adjusted the output current such that it would hit the voltage set point assuming an optimal loudspeaker impedance, this charging would cause a lag in the output and the voltage rise or slew rate would be fairly slow. Instead, it opens up the current well beyond what is necessary to maintain the set point which allows the driver and crossover components to charge very quickly. Once the set point is reached (or preferable before to avoid overshoot), the current is backed off such that the voltage across the loudspeaker once again matches the set point. An amplifier that can source a lot of current during that surge means that it can maintain a very fast slew rate even with a large amount of inductance and capacitance within the loudspeaker. This mainly shows up in highly dynamic sources where the input signal has large, sudden swings. A high-current amp will be able to change the output quickly enough to accurately reproduce the large swings in the input while low current amps will potentially cut off some of the sharpness of the signal.
What you are probably thinking of, is "dynamic headroom." The ability to give a brief blast of extra power when called for. An amplifier with a very heavy duty power supply. A power supply having power (current) in reserve, the ability to exceed the specs. A large transformer and plenty of filter capacitors.
If you have an amplifier that can put out 150 watts into 8 ohms, you know that it can deliver 4.3 amps of current. If you moved up to a 250 watt amplifier, it can do 5.6 amps. You'll likely won't need 250 watts of constant power at normal listening volumes, but when a large, sudden bass note happens, the ability to deliver extra current will make the leading edge of that note much more sharp and accurate.
