UVA radiation penetrates the outer layer of your skin to your dermis, where it degrades the skin's supportive proteins. This leads to a gradual loss of elasticity, and the development of fine lines and wrinkles, as well as as photo-induced hyper-pigmentation, or 'age spots'. UVA is also considered responsible for sun allergies.
UVB reaches the dermis to a lesser extent, because it wreaks its own form of havoc in your epidermis, the outer layer of skin. Here it literally burns skin tissue, causing the reddening, blistering and pain we associate with sunburn. It is thought that isolated doses of UVB radiation (on your annual holiday for example) over a number of years is fundamental to the development of skin cancers, including malignant melanoma.