What It Is

The above graph shows a very averaged and idealized cross section of the atmosphere, with a very visible jet stream being that red dot. This requires some explanation, though, since the scale is weird (well, atmospheric scientists don't find it weird, I guess). What you see is a slice of the atmosphere. Picture a cake. Or, instead of picturing it, just look at one:
Cake.
Good. Looking at it from the top, you can see only icing and some nuts; if
you want to see the layers and what's inside, you have to cut it and look at it
from the side, like in the above picture. That's called a cross section,
and in the graph above, you'll find a cross section of the atmosphere, kind of.
The cross section goes from the equator to the pole. How is this done?
Imagine this (you'll actually have to imagine this one). Take a very sharp
knife and make a cut from the equator to the North Pole, just cutting straight
north, through the atmosphere, and remove a section, like you'd take a slice of
cake, and look at what's left from the side. What's pictured in the graph
is the wind speed, in meters per second, through the cross section. Kinda.
The real wind speed on any given day doesn't look like that (you'll get to see
it later, don't worry), so the wind speeds for every day between 1968 and 1996
are averaged, and that's what you see. Kinda. Actually, when you
look at a slice of cake from the side, you can only see that one cross section,
so every cross section from the equator to the pole, along every meridian, was
averaged together to form that image. That's what it really is. Here
it is again:

Another snag is that this image isn't drawn to scale at all. It looks like the atmosphere is very tall, and the planes fly very high, and other such nonsense, but actually, the atmosphere is REALLY thin. The jet stream is actually at around 12 km high. That's the center of that red dot, around. 12 km. That's about 40,000 ft. Let's compare it with the length of the graph. The radius of the Earth is approximately 6,400 km, so from the equator to the pole, the distance is about 10,000 km (you can check my math if you feel like it; it's just a quarter of a circumference). Let's compare them, shall we? Height (to the red dot): 12 km. Length: 10,000 km. That's a factor of almost 1000, though the picture clearly doesn't have those dimensions! The atmosphere is thinner, relatively speaking, than the thin candy shell that makes an M&M melt in your mouth rather than in your hand! And yet all our weather happens pretty much within those 12 km. The atmosphere actually goes a lot higher than that, up to 100 km and beyond, but the interesting part -- the part that has weather -- is pretty much those 12 km and not much more.
I wasn't joking when I said it wasn't drawn to scale. See those numbers on the side? (No? That's because they're small. Try looking at the picture at the top of the page.) They actually represent pressure rather than height. Pressure is essentially the weight of everything above you per unit area, and it's measured in millibars, abbreviated mb (by atmospheric scientists, at least; other people use other units, like the pascal, the torricelli, and the psi -- pound per square inch). One millibar is 1/100 of a kg/(m*s^2), and what we call an atm, "standard" atmospheric pressure at sea level, is 1013.25 mb. That's, again, a measure of the weight of the atmosphere above you. Underwater, the same thing happens; the deeper you go, the more water there is above you, and the higher the pressure. In the atmosphere, the higher up you go, the less air there is above you, and the lower the pressure. There's a catch, though. Air doesn't have a constant density (mass per unit volume, or, essentially, how much "stuff" there is in a given space). As the pressure decreases, so does the density. (Remember the ideal gas law? No? That's OK.) The end result is that those numbers on the side that seem pretty equally spaced on the graph are actually REALLY not shaped that way in real life. The distance between the numbers at the top is much greater than the distance between the numbers at the bottom! To compare, the 200 mb level is at 12 km, as I said before, but 100 mb is at 16 km. That's a difference of 4 km. 1000 mb is at about 0 km (sea level), and 850 mb is at about 1.5 km. That's a difference of 1.5 km for a larger difference in pressure.
SO WHAT IS IT?
It's
a tube of very strong westerly wind that hangs out about 12 km high at about 40
degrees latitude (N and S). It's not a tube by itself; clearly, from the
graph, the change is gradual. This tube meanders around quite a bit,
always blowing in the same direction -- west to east. (Westerly means FROM
the west and TO the east, which, however silly you might find it, is the way
atmospheric scientists do things. You'll notice from the graph that the
winds near the equator are easterly. Those are known as trade winds, and
you can look them up later if you like.) The winds are high up, as has
been pointed out repeatedly, and as you can see from the graph, they aren't very
strong at the surface, so buildings don't get blown away by the very strong
westerly wind that hangs out about 12 km high at about 40 degrees latitude (N
and S). That's what the jet stream is, in a nutshell.