First of all, what is coriolis force?
Picture a turntable turning anticlockwise as pictured from the top. Now consider an ant crawling out from the centre of the turntable to the edge of it, in, what looks like the ant to be, a straight line.
Now look at the path the ant travelled form the point of view of an observer looking down on the turntable from an inertial reference frame (i.e. standing still!). The ant actually travelled on a path that is curved to the left. In order to do that, the ant had to be accelerated to the left. It did that by hanging on tightly to the surface of the turntable, which provided a force to the left.
(Actually, if the ant walks in from the edge, or in any other direction that is a straight line with respect to the turntable, he also curves left according to the outside observer.)
Standing as that observer, it's obvious what happened. A force was applied to the ant, and it accelerated in a curved path. But sometimes it's really awkward to stand away from the rotating frame like that. So to work out what happens to the ant more conveniently, we invent a ficticious force called the
coriolis force, which always acts to the
right when the ant tries to move. As far as the ant is concerned, the force the turntable applies to the left just balances the coriolis force to the right, and he travels in a straight line. That's much easier for the ant than trying to picture the scene from the point of view of the observer! The coriolis force corrects for the fact that the turntable is actually spinning, allowing the ant to use the turntable as its frame of reference.
We do exactly the same on the surface of the earth, when we don't want to solve the mechanics by blasting off into space and looking at the situation from there. We invent a coriolis force, which acts to the right when something moves across the earth (in the N Hemisphere). That accounts for the fact that the earth is not still, but actually spinning. The formulas are slightly more complicated in 3D.
Why doesn't it affect aircraft?
It does, but it's tiny. Even for something travelling at 600 knots, directly towards the axis of the Earth, the coriolis force is about 0.002 G. Compared with all the other forces on the aircraft, that's negligible.
It's only important when you consider things that
only have small forces acting on them. A parcel of air in the atmosphere is affected by pressure gradients. Take a sheet of wood half an inch thick, and point the face towards the centre of a low pressure area. Did you feel it pushed towards the centre of the low pressure? No, I thought not.
Yet it's that pressure gradient that is balancing the coriolis force for the wind aloft. Both forces are very small, of the order of a ten-thousandth of a G. Only with something like air does it become important.
Hope that helps.