I suspect the concept you are missing is "output impedance". A voltage amplifier has some voltage gain but probably a high output impedance. When it is connected to no load, it shows a nice big output signal but as soon as you attach a load, the signal disappears.
The output impedance and load impedance act together as a voltage divider.
For example, you might have a voltage amplifier producing 20V open-circuit and it has a 1k output impedance. Connected to a 100k load, the output voltage will be 20*100k/(101k) = 19.8V... but when you connect it to a 10R load, the output voltage will drop to 20*10/(1010)=0.198V.
simulate this circuit – Schematic created using CircuitLab
If you want to get some power (voltage * current) into the load, you need a high voltage output with a low output impedance so that it is capable of supplying current and therefore power to the load.
In a classic audio amplifier, the output stage is NOT a "current amplifier" but a low-impedance voltage follower. "Current amplifier" has the implication that its output is some linear multiple of its input current, and that's not quite what's happening. The purpose of the final stage is to provide enough current to the load so that the load's voltage follows the open-circuit voltage of the amplifier (to grossly oversimplify).
Feedback is also important. The amplifier uses feedback to correct its internal nonlinearities and the voltage-divider effect of its output impedance interacting with the load impedance.
As the load impedance goes lower, the amplifier dissipates more power/heat. Maximum power-transfer into the load (for a given output impedance) is when the load and output impedances are equal, but that is a really bad situation for an amplifier because it will have insufficient damping factor. Normally you want to keep the output impedance well below the load impedance in order to maintain stability of the amplifier's feedback loop and to avoid the amplifier getting unnecessarily hot.
Edit for the ignorance and downvote fairies out there. Here's a classic and famous audio amplifier design, from Doug Self.
Note the three stages: an input differential stage which is used to implement feedback, a voltage-gain stage in the middle which has an output impedance of a couple hundred ohms, and finally an emitter-follower (low-output-impedance voltage follower). There is voltage negative feedback from the output back to the input differential, resulting an an overall closed-loop amplifier voltage gain of about 20.
Summary: the amplifier as a whole implements a voltage source with low output impedance. If you put an 8 ohm load on it, the load will get about twice as much current and therefore twice as much power as a 16 ohm load.