A high-current amplifier is simply one in which the power supply and output stage can pass enough current to drive low-impedance loads. The "high-current" label is pretty much a marketing label, it gets tacked onto even entry-level receivers with slightly beefier power supplies than one usually sees at that level, but not much else (there is still only one pair of output transistors per channel, driving low impedance loads with a single pair can push the transistors out of their safe operating area - this is why many receivers have a "4-ohm" mode, it limits the supply rails and engages current limiting in the output stage to keep from blowing outputs.) When an amplifier attempts to swing a given voltage into the load, the load will attempt to draw a certain amount of current, and if the power supply cannot provide it or the output transistors cannot safely pass that current, output either sags (and the amplifier clips, and you can toast your tweeters if the distortion harmonics are strong enough) or you toast outputs (and if the protection circuit doesn't trip, possibly your woofers as well since up to 30-50VDC could pass through whichever transistor shorted to the speakers.)
To deliver large amounts of current, you need a beefy power supply, multiple output device pairs, and GOOD heat sinking. The amplifier can be class A, AB, B, D, whatever...the physics still apply. Also, you really can't divorce power in watts from current in amperes, since P = IE (P is power in watts, I is current in amperes, E is potential in volts.) If an amplifier cannot deliver the current a speaker asks for, the power in watts dissipated into the load will not be there either.