If you were to distill down an amplifier circuit to a basic "triangle" block diagram, there are two inputs, one inverting and one non-inverting . . . and one output. The amplifier derives its output from the difference between the two inputs. A perfect, ideal amplifier would have infinite gain at all frequencies, thus the actual circuit gain is set by the feedback network between the output and the input(s).
Most audio amplifier circuits (power amps, preamp stages, phono stages, etc.) are "voltage feedback", that is, the amplifier output is based on the voltage difference between the inputs. Since most amplifiers (that are modeled in this manner) have a very high gain, you can think of it in the sense that the amplifier circuit is always working as hard as it can to bring the voltage between the inputs to zero.
A "current feedback" amplifier is one that derives its output from the current flowing between the two inputs, rather than the voltage . . . that is, it's always trying to bring the input "error current" to zero, rather than the input "error voltage". In the simplified world of ideal, perfect amplifiers, there's really very little difference between the two. This is because when you add resistors to form a feedback network, the voltage and current differnce tend to be the same thing due to Ohm's law.
The differnce between the two topologies is only apparent in implementation, because in the real world, amplifier circuits don't have infinite gain and bandwidth. The main difference to the engineer is that in the voltage-feedback design, gain and bandwith are always linked together, and one must frequently choose between them. The current-feedback design has bandwidth linked to output/feedback loop current . . . and because of this, the gain and bandwidth can be much more independent of each other.
Just as everything in life, and moreso in audio, neither is "better", they're just different weapons, if you will. Voltage-feedback designs typically have better DC performance, and lower noise. They're also much better suited to designing active filters. Current-feedback designs allow the engineer to acheive higher slew rate and bandwidth for a given gain, which usually means a little lower distortion. The output load current must be held to a smaller range, however.