Hi Mark,
Sunspots have quite a complicated movement. Roughly speaking they move with the rest of the Sun, rotating once every 27 days or so. But when a sunspot pair emerges the two spots can also have twisting movements, and then the two spots slowly separate, and also it appears that they go just a little bit slower than the solar surface around them, but that is hard to see.

The reason that they have a dark core (umbra) and an outer lighter core (penumbra) is that both are colder than the rest of the sun, but the dark inner core is colder than the light outer core. The surface of the sun is about 6,000 degrees centigrade, and the centre of a sunspot is about 4,000 degrees centigrate. The outer ring of the sunspot is somewhere inbetween. In fact, both of these only look dark in contrast to the rest of the Sun. If you could cut out the rest of the sun and only look at the spot it would certainly be shining bright enough to see.

Now, why the spot is cooler than its surrounding is another question altogether....