At a given impedance, say 8 ohms, the current of the amplifier is directly related to the specified wattage. If your speakers are rated at a flat 8 ohms across the frequency spectrum, more watts equals more amps, i.e. current.
The issue becomes more complicated when a speaker has a widely varying impedance rating across the frequency spectrum. For example, my Thiel CS6 speakers range from about 2 ohms up to around 8 ohms depending on the frequency. In order for the amplifier to supply the necessary power to allow the speaker to have a flat frequency response it needs to double its output with each halving of the impedance. My Krell KSA 300S outputs 300 watts at 8 ohms, 600 watts at 4 ohms, 1200 watts at 2 ohms, and 2400 watts at 1 ohm. When people talk about a high current amplifier this is what they mean.
Current relates to amps. The more amps your amplifier produces the more "current" it has. You can calculate the amperage of your amplifier from the wattage specs at the rated impedance at those wattages. Amps are calculated by dividing Watts by Volts. 300 watts divided by 120 volts = 2.5 amps. 2400 watts divided by 120 volts = 20 amps. Big difference.
If an amplifier outputs 500 watts at 8 ohms and 500 watts at 4 ohms it would not be considered a high current amplifier - at least in my world. It doesn't matter how big the power supply is or how expensive the parts are. It simply cannot output the amps that a speaker with a demanding impedance curve requires to produce a flat frequency response. There's nothing wrong with that, however, if you are running a speaker with a flat impedance curve and the amp has the watts to drive the speaker to an acceptably loud level. Devore speakers, for example, are designed to have a flat impedance curve so they can be driven by a wide range of amplifiers that don't produce high current.
People get wrapped around the axle about "current" but most speakers present a fairly benign load and do not require the heroic output of a Krell, Pass, D'Agostino, or Levinson amplifier. But if you are running certain Wilson Audio, Thiel, Apogee, or other speakers that present a difficult load then a high current amp is necessary for them to sound their best.
Sorry for the long post. I hope this helps.