When would the appropriate time or scenario be to use different metering systems - spot metering, reflective or incident? And why? I do not understand when to use these.
Let's start with some definitions:
This is where you meter the light being reflected by the subject. This reading can be taken by a dedicated light meter or by the one built into your camera.
This is a type of reflective metering where you meter the amount of light being reflected by a narrowly measured range. This range, when measured through your camera, changes based on the lens. Spotmetering.com reports a Canon 5d as using a 7.11 degree angle of measure when using a 50mm lens.
In this case, your camera would be measuring whatever is being reflected within a 7.11 degree angle, from wherever you're standing to the object reflecting light.
A dedicated light meter, on the other hand, can have an angle of measure as small as 1 degree.
This is where you use a dedicated light meter to measure the amount of light falling on a subject (as opposed to being reflected by a subject)
When would the appropriate time or scenario be to use different metering systems?
The "truest" or most accurate measurement will be made with an incident light meter. Reflected metering tends to measure everything assuming you want middle gray. While this works in many situations, it tends to error when photographing either very bright things (causing them to underexpose) or very dark things (causing overexposure). You can test this by metering a white shirt. Shoot a "proper" exposure. Now manually force the meter to read +2 and shoot again. The "overexposed" shot will be a better exposure.
You could also compare the meter reading from an incident meter placed next to your shirt and compare it to what your camera says.
Given this, why would anyone ever not use incident metering? The answer is fairly simple: you can't get to your subject. Maybe your subject is a grizzly bear, maybe it's a soccer player. In either case you will need to use a reflected meter.
Your camera probably has Evaluative, Center Weighted, and Spot metering modes.
So why use spot?
You want to use spot metering when the scene is highly contrasted or backlit - in situations where the exposure is going to vary wildly across the scene and so you want to make sure that your actual subject is exposed properly.
Why use the spot meter on a dedicated light meter vs. what's in your camera?
Honestly, I wouldn't. The exposure difference will be more accurate with the dedicated light meter - but it will probably be very close to what the camera says. In these situations, I find it easier to take a shot and check the histogram as opposed to getting the light meter out of the bag,