What is our skin made up of?
The human skin is indeed made off a number of different layers, the three most important are epidermis, dermis and hypodermis (also called subcutaneous fat), see the figure (from here) for details:



All three layers can be subcategorized further, I'll only give a few details here. More can be found for example in the Wikipedia article on skin.

Epidermis: Top layer of our skin, is covered with the stratum corneum, which consists of several layers of dead cells, which protect the layers below. New cells of the epidermis are made at the border between dermis and epidermis and travel upwards (actually the top layers are worn off). The epidermis is the place where the pigmentation takes place due to melanocytes embedded in the keratinocytes of the epidermis. 

Dermis: The next layer is the dermis where a lot of important functions on the skin happen. Sweat and oil are produced in the dermis. The feeling of the skin is also mediated in the dermis where the ends of the nerves are embedded. The hair roots of the hairs of the skin are also located here as well as the blood capillaries which supply the skin.

Hypodermis or Subcutaneous fat: The last layer is the hypodermis. It attaches the dermis to the underlaying muscles and bones and houses bigger blood vessels and nerves which go up into the dermis. The fat layer is also important for controlling the body temperature and storing fat reserves.