What exactly does a FET do?
What does a FET actually do?

A FET is a transistor which uses the field effect to control current flow.
The trick is that applying a (constant) electric field does not need any current and does not require a flow of energy (power) to maintain that field. So when using a FET you only need a voltage to control the flow of current in a steady (DC) situation.
Current is only needed when you want to change that voltage. The input of a FET behaves like a capacitor, you need current to charge/discharge it but once the voltage you want has been reached, no more current is needed. This is true for JFETs and MOSFETs.
There are depletion type FETs, these are &quot;normally on&quot;, when their gate-source voltage \$V_{GS}\$ is zero, current can still flow. Nearly all JFETS are of this type. Depletion MOSFETs exist but are rare.
There are also enhancement type FETs, these are &quot;normally off&quot;, when their gate-source voltage \$V_{GS}\$ is zero, no current can flow. Only when a certain \$V_{GS}\$ is applied, current can flow. Most MOSFETs are of this type.
N-channel MOSFETs need a positive \$V_{GS}\$ to allow current flow (switch on).
P-channel MOSFETs need a negative \$V_{GS}\$ to allow current flow (switch on).

What I really want to know is what happens when current is supplied to the gate of an N-channel FET and a P-channel FET.

As I wrote: these devices are voltage controlled so you don't &quot;supply a current&quot;. Instead, you apply a voltage.

If I were to connect an LED...

Go search for how to control an LED with a MOSFET, there are plenty of examples that show how to do this. With the information above you should be able to figure out how this works.

What would the difference be between PNP/NPN transistors and N/P-channel FETs?

Bipolar transistors (NPN, PNP transistors) have a very different working principle. These devices do need a current to control the current flow.
Both MOSFETs and bipolar transistors can be used to switch on/off LEDs. In general when using a bipolar transistor, you need a resistor to limit the base current and you need to supply a (small) current to the transistor. When using a MOSFET a resistor in series with the gate is generally not needed as no current will flow anyway. Some MOSFETs do need a large \$V_{GS}\$ (several volts)to fully turn on, that is sometimes an issue.

If my initial ideas about what a FET does were right then surely a FET is basically the same as a PNP/NPN transistor...

Yes/No. Both devices are VERY different. However, both can be used to switch a load. Both can be used as a signal amplifier. The fact that both bipolar transistors and MOSFETs are still used in large quantities means both types have their place. JFETs are used much less though, only in certain niche applications.
I suggest that you look at example circuits that show how all these transistors are used. You don't have to fully understand these devices to be able to use them.