The winter solstice is the boundary between fall and winter--all the seasons begin and end as astronomical events, rather than changes in the weather.
Recall that all globes come with a built-in 23 1/2 degree tilt. That tilt represents the degree to which earth's axis of rotation is out of perpendicular to its orbit around the sun--as it would be if you put a (small) model of the sun on the same table with the globe. Model our seasons in the following way. Put a powerful lamp in the middle of a large table. Put the globe on the edge of that table with the north pole tilted as far as it can go toward the sun: it is June 21st, the summer solstice. As you spin the globe, notice how high the sun would get in the imagined sky of North America; notice also that much more than half the northern hemisphere is illuminated. (It's hard to see this with the usual shiny-surfaced globe; I dust it with chalk dust to make clearer how much is lit.) Because of this, the northern hemisphere day will be longer than its night, and the sun will shine more directly (so more intensely) on the surface at noon. Both of these factors mean more heating of the northern hemisphere, bringing on warmer weather as the heat builds up.
Now slide the globe counter-clockwise around the edge of the table to make the weeks and months pass. While you do, be careful to keep the axis of the globe point in the same direction all the time (keep it aimed always at the same side of the room). When you reach the opposite side of the table, you will find the north pole now tilted away from the sun. It is now Decenber 21 and the winter solstice. Notice that the situation is reversed from six months ago: the sun will not rise nearly as high for North America, and since most of the hemisphere is in darkness, the daytime will be short. The earth's surface in the northern hemisphere is receiving much less heat than six months ago, so it is getting colder.
This might put a few misconceptions to rest. First, notice that our distance from the sun is not a big factor. In fact, the earth is actually a bit closer to the sun right now than it will be in June! Notice also that the seasons will be opposite in the two hemispheres: summer has just begun for my friends in New Zealand, who are enjoying their longest days right now. Notice that the tilt of the earth'a axis is not changing--just where the sun is in relation to it. [You might remember that the north pole always points (approximately) at Polaris, the "pole star."] Finally, don't confuse orientation with distance: plenty of students do this very activity, then get mixed up explaining that when the north pole is tilted toward the sun that hemisphere is actually closer to the sun--true, sort of, but only by a miniscule fraction of a percent!
Remember: the changing seasons are about the length of days and the directness of rays!
One question might still occur to you: why isn't the winter sostice the coldest day, and the summer solstice the warmest? It's strange to think that, as the days lengthen into January, the weather is still cooling off! The secret is to think in terms of the balance between heat gain and loss: the summer solstice is when the northern hemisphere gains heat fastest, but it takes time for that temperature to rise; similarly, the decreased heating that reaches its lowest ebb on Dec 21 will take time to have its full effect. So it really does make sense to begin winter with the winter solstice: though the fastest cooling is past, the chill will still deepen further.