The warmest types of socks are either manufactured out of wool or synthetic materials that are designed to provide insulation. Design-wise, mountaineering socks are usually the warmest option because of their thick material. They are intended for long periods of wear in cold temperatures.
Wool socks are popular due to their warmth and ability to keep the feet warm even when they get wet. Some wearers may find them itchy, but blends that incorporate merino wool are not likely to cause this issue.
Socks made of synthetic insulating materials are able to hold in heat like wool, but they have quicker drying times and are less likely to rub against the skin.