Well, you know that the earth's axis is tilted about 23°, right? Well, the vernal and autumnal equinoces are on the 21st of March and September, respectively; the summer solstice, on the 21st of June, is nominally the longest day of the year in the northern hemisphere, and the winter solstice, on the 21st of December, is the shortest.
This is all caused by the tilt of the Earth, and thus the southern hemisphere is exactly reversed; just like Hawaii midnight is at the same time as GMT midday, South African winter is at the same time as British summer, and vice versa.
Thus, by the strictest possible definition, winter is defined as the coldest season of the year rather than by a month-based definition, but the northern winter is usually considered to be "the winter".
It all gets rather complicated, I'm half asleep, and I'm just going from memory, so I may not be entirely correct, and I've definitely been very unclear, but it'll have to do.