Evaluative is the clever one (Nikon's matrix). It takes dozens of readings from all over the image. In that way, it not only knows what the average scene brightness is, it knows the values of the brightest and darkest tones, it knows where they are, and it knows how large an area they occupy.
It's also linked to the AF system so it knows where the main subject is (if you're on auto AF select) and how far away it is. It then interrogates this data and applies adjustments when necessary, for example if there's a large area of bright tone at the top, and a distant AF setting, chances are that's sky in landscape so it will down-weight the sky to prevent under exposure.
Evaluative will almost always get you close, if not exactly right. Good default mode.
Spot is also handy, if you know what you're doing and want to reference a particular area. Only use strictly in manual, or the exposure will be all over the place like a rabbit in a field of wet turnips.
Partial is a bigger spot - not sure what that's useful for.
Centre-weighted reads the whole image with an emphasis on the middle. In practise it's similar to evaluative, but without the smart thinking bit, so you always know where you are and the camera is less likely to do something unexpected. Sometimes favourite for more consistent E-TTL flash exposure, which can get very complicated and throw the camera's little evaluative brain into doing something silly.
At the end of the day, all metering methods are calculated guesswork. If you want the most accurate exposure, check the LCD and learn to read the histogram with blinkies enabled (highlight alert). That shows an actual recorded image, so cannot be wrong. Tweak and adjust from there.