The thickness of skin on the nose is one of the least changeable parts of your nose. Thick skin will result in a nose that has softer angles, less definition, and is broader. Thicker skin will also retain swelling longer after surgery. Some surgeons will attempt to thin nasal skin, however I do not agree with this practice. It carries a high risk of complication which may permanently disfigure you.
Thicker nasal skin is more common in darker skin types. Commonly, thicker skin will have larger pores and be more oily. You can also get a good idea how thick the skin is by touch. If it is easy to feel details of the bone and cartilage then it is likely thinner. Thinner skin also tends to be more easily slid over the underlying tissues.
Generally, thick skin is only an issue on the tip. While it is possible to have thick skin on the upper part of the nose, it generally does not challenge the result of a rhinoplasty like thick skin at the tip.
Nasal skin may get thicker as you get older. This is most true in people with rosacia.