how does current work in an amplifier?

A formula that might help is the following:

Current in Amps SQUARED = Power in Watts DIVIDED by Resistance in Ohms

For comaprison:
100 watts into an 8 ohm speaker requires 3.54 Amps
100 watts into a 4 ohm speaker requires 5.00 Amps

Not sure if this helps at all, but...

I can't give you an in depth explanation. But if you look at the listed specifications for any given amp you should see the Amperes (Amps) that the Amplifier is capable of delivering. You should find a specification for both continuous and the peak number of amps the amp can produce.
The cut off, delineating a high current from an ordinary (non high current) amps has not been officially established AFAIK. But you can do your own informal survey of the big amps with reputations for handling demanding loads and following those specs as the amplifiers get progressively more powerful, to get a general idea of the range you are considering.
My own completely arbitrary idea of the lowest limit for a "high Current Amp" is about 20-30 amps continuous. but be aware their is alot more to it than this single figure.
I am prepared for the onslaught of scientific explanations to follow which will point out the critical importance of other specs to consider. Mine is a simplistic view.
BTW a really big amp will put out over 100amps non peak just so you know. Pay some attention to the slew rate as well which tells you how fast the amp can deliver that power when needed.

The math is easily obtained, yet a little 'vague' in terms of practical application by the typical user of audio goods.
That's why I tried to use a real world example, certainly a more difficult to write, yet more graphic representation of the answer, hopefully, to the question of the posting a'gon member.

KISS method, lol.

Larry

A high current amplifier has a very large transformer with a compliment of large capacitors. A SS amp has an array of semiconductors (transistors or mosfets) that are either cascaded or cascoded. These devices are connected to the transformer and capacitors and function as variable resistors to create a large, powerful signal that is identical to the small input signal coming in from the preamp. That large, powerful signal drives the speakers. The Capacitors and Transformer are energy storage devices. They provide the peak power needed as the music signal varies from low to high. Imagine having a big rheostat that ranges from -60 to +60 volts that is connected to a speaker. If you could turn the dial back and forth fast enough to mimic a musical signal- you could make music. A tube amp functions much the same way. The tubes get the preamp signal and vary their high level output voltage to mimic the musical signal. The output of most, but not all tube amps goes through transformers. That limits their current carrying capability and that is why tube amps generally have the same power rating for 16, 8 and 4 ohms. SS amps can output higher power levels into lower impedance speakers up to the point that the semiconductor devices overheat and go up in smoke. To demonstrate, apply a dead short across your SS amplifier. (Although many modern SS amps have overload protection devices which kind of takes the fun away). Very simplistic terms, but that is how any amplifier works.