Dehydration is a big part of it, but so is the general breakdown of alcohol in the body. The alcohol breaks down into some harsh compounds like acetaldehyde and formaldehyde, then it breaks down into acetic acid.
Acetic acid is fine, it's the same stuff in vinegar.. the problem is the middle part. The common reaction to those two is sweating, skin flushing, vomiting, and nausea.
Combine it with water loss from drinking other things than water, throwing up, or peeing more, and then add in the two compounds and it's a painful morning.

A hangover is basically moderate dehydration and all the symptoms associated with it (muscle fatigue, headache, slight nausea). This is caused by the liver trying to process and get rid of the alcohol you consume. One of the most efficient ways to do this is to break the sugars in the alcohol down and pass it out of the body as urine. So the more you drink the more dehydrated you actually become.

Your liver produces an enzyme called alcohol dehydrogenase, which it uses to "detoxify" ethanol. Unfortunately it breaks down ethanol by converting it to ethanaldehyde. The symptoms of ethanaldehyde poisoning are nausea, vomiting and headache.

Your liver also has a trick for converting ethanaldehyde into ethanoic acid, but if you drink enough you're going to have all the elements of the ethanol - ethanaldehyde - ethanoic acid chain floating around in your blood at once, because it takes time to process it all.

The reason people end up blind, paralyzed and dying from drinking methanol is that your liver tries to process that with alcohol dehydrogenase too, and converts it into methanaldehyde, aka formaldehyde, which is quite toxic and especially harmful to nerve cells.

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