Let's start by defining some terms. There are two kinds of milk, pasteurized milk and 'raw milk'. Raw milk is the milk as it comes out of the cow (although the cream is usually removed). In many parts of America it is illegal to sell raw milk because it is too delicious and healthy it can carry disease. In most countries the milk that you buy at the supermarket will be pasteurized, or even ultrapasteurized milk. There are downsides to consuming pasteurized milks, but that's another node.
Most of us have experienced pasteurized milk that has gone sour, and wish that we hadn't. It is essentially rotten milk, and it is undeniably disgusting and inedible. It will make you sick. Needless to say, that is not what this node is about.
Some of us have had the fortune to be introduced to raw milk that has gone sour, preferably with a known and delicious culture of healthy bacteria. This milk is healthy and (yes really) delicious. The most common cultures are of the genus Lactobacillus, a probiotic that can improve the health of the human gastrointestinal tract, and which are also found in products like kefir and yogurt. Americans would generally recognize all forms of sour milk as 'buttermilk', although if you are lucky you may also find them under the labels fermented milk or cultured milk. Sour milk can have many different flavors, but all types are tangy and strong and sour; some are sweeter, some are more 'fermented' and tangy; in my experience, most are not as strong and thick as the buttermilk found in American supermarkets.
There are still many parts of the world in which sour milk is the norm, and you have to specify 'sweet milk' if that's what you want (and it may be mixed up from powder unless your hosts have recently done the milking). A good culture can start souring milk in a matter of hours, and it is not uncommon to add the remains of the last batch of sour milk into the sweet milk as soon as you're done milking. The benefits of sour milk are many, and do indeed include taste. In areas without refrigerators, sour milk can be stored at room temperature for a couple days with no ill effects, and its life may be extended even more if you have a cool place in which to store it. And perhaps most importantly, because the bacteria break down the lactose, it is easier to digest than is sweet milk.
The downside to non-pasteurized sour milk is that any bacteria present in the milk, which might include some big nasties like tuberculosis, diphtheria, and typhoid, are still present in the soured milk. If you have healthy and vaccinated cows and milk them in a sanitary fashion there is not much of a risk of catching anything, but you do have to trust your milk source.
Sour milk is good plain, on cornbread (dried or fresh), in grits or any sort of hot breakfast cereal, and, if all else fails, over dry biscuits. If you have a sweet tooth or are a bit queasy when faced with sour milk, you can add sugar (I highly recommend at least trying sugar and sour milk over dry cornbread). There are thousands of regional recipes that use sour milk in any way that you can imagine; I recommend trying as many as you can.
Sour milk may also refer to sweet milk that has been 'soured' with addition of an acid, usually citric acid (in the form of lemon juice, for example) or acetic acid (in the form of vinegar). If you need sour milk for baking, it does not matter what kind of sour milk you use, and you can 'sour' milk by adding one tablespoon of the aforementioned acidic substances to one cup of milk.