The most important job socks do is prevent blisters. I suffer from a skin condition ("non-Hallopeau-Siemens recessive dystrophic epidermolysis bullosa") that makes my skin blister easily, so I have looked carefully at socks and tried a number of different types.
The unholy trinity of blisters
Three factors work together to cause blisters; heat, moisture and friction. Socks don't do much to reduce heat, but they do reduce friction and moisture.
Don't use cotton socks - cotton is one of the worst materials that you can choose for socks (or any athletic wear). I generally use Cool Max, which is a synthetic fiber that uses capillary action to spread out sweat so it evaporates better. There are many other synthetic materials on the market that also work well. Some people get on well with wool; I use wool socks for hiking in cool weather, but not for running. If you choose wool, get high quality merino Wool, such as 'smart wool'. I've tried Bamboo socks; they feel nicer than Cool Max when dry, but hold more water and dry more slowly. Some socks include silver as an antibacterial material, which helps reduce odor. This is great in socks where you can't wash them for days at a time (long backpacking trips), but for running it is not a big deal. If you have a problem with shoe odor, then silver based socks may help.
Modern construction techniques allow for seamless, or near seamless construction, which is useful, as seams often cause blisters. These construction techniques also allow for variable thickness socks; I've not found much benefit from this approach.
If you are wearing light weight racing shoes for a short distance race, then go for light weight socks. Otherwise, I would not worry too much about sock weight.
I have not experimented extensively with compression socks. Compression over the calf does seem to help recovery, but compression on the foot itself causes me problems. Compressing the bones in the front of the foot (metatarsals) can put pressure on the nerves, and the nerves do not appreciate this (Morton's neuroma).
If you ever have a problem with blisters on your toes, try Injini socks. They have individual sections for each toe; they are like a glove for your feet instead of a mitten. They take a little getting used to as your toes are not used to being separated. Having adapted to Injinji, I really like them. They make both a normal thickness sock and a much thinner 'liner' sock. The liner can be hard to find; Zombie Runner is the only place I could find them. The liner sock is thin enough to allow them to be worn with other socks as well.
Double layer socks
The idea of double layer socks is that the extra layer of material reduces friction. The inner layer rubs on the outer layer, rather than rub on the skin of the foot. I like the Wrightsock dual layer socks. I generally wear their thinnest Cool Max sock, though I also have some of their thicker Silver based socks.
Injinji + Wrightsock
My favorite combination is to wear the Injinji liner sock with the thinnest Cool Max Wrightsock. This gives me toe protection and three layers of material. I find that this pairing is much better with blisters than any other option I've tried.
Tangent - Fit is critical
The most important issue with blisters is having shoes that fit correctly. The combination of socks I use allows me to swap for slightly thicker socks to tweak the fit.