A closeup of the skin near my knuckle.
Photo by Matthew T. Ross

The skin of animals is an incredibly complicated structure containing many layers of tissue. Skin contains hair follicles, sweat glands, sensory cells, and all sorts of other structures. The most important properties of skin is that it is tough and yet flexible. The stretchiness comes from collagen fibers in the dermis (the lower tissue layer of the skin). Alternating areas of thicker and thinner skin also help to make the skin flexible. 

Above you can see an image of my knuckle. The larger grooves help with flexibility and prevent too much skin from bunching up at joints (such as a knuckle). The lines on your hand are a great example of this. The lines on your hands show up at about week 12 of gestation.

Wrinkling due to aging seems to be largely caused by fragmentation of collagen proteins. Xia et al. published a paper this year (2015) looking at what causes collagen fragmentation in a mouse model of aging. You can find that paper here: http://www.jdsjournal.com/article/S0923-1811(15)00034-1/abstract