The phenomenon that causes stars to twinkle is atmospheric refraction. That is, the air above us is a big ocean of not very dense fluid that has a small bending effect on light that passes through it. If it were perfectly still, the light from a star would follow a path that stays put, and the star's image would appear stable.
But the atmosphere is always moving; winds in every level of it cause slight density differences to move around, and different densities bend light by different amounts. And so the light rays from a star zig and zag as they head toward your eyes. That makes them appear to twinkle. The reason this happens for stars and not planets or other larger, "extended" objects, though, is that a star's apparent (angular) size is so very, very tiny. It is what is called in optics, a "point source."
When there's a little bit more of an apparent disk (these are measured in angle, and we're talking in the arc-second range and smaller), then it takes a lot more bending of the incoming light for the rays coming from all the parts of that disk to get bent away from your direction so that the (extended) object would appear to "blink" momentarily, and so, appear to twinkle. And it doesn't take much of an apparent disk to prevent twinkling.
But it also depends on what's happening in the air above you at any particular time, too. On some clear nights, there will be a lot of roiling in the atmosphere, and stars will twinkle more. Other clear nights, when the upper air is calm, twinkling will all but stop.
The whole effect is sometimes likened to what you see on the bottom of a swimming pool in bright sunlight. If the surface of the water is pretty wavy, there will be a lot of moving light and dark patterns on that pool floor. If nobody's been in the water for a while, and there's no wind, the surface gets smooth, and the moving patterns disappear.
And it has nothing to do with the vast distances, or anything between the stars and the top of our atmosphere, because there's virtually nothing in that space but a very hard vacuum. Wikipedia has a nice, concise treatment of the effect. (not, "affect"!) But their explanation that planets don't twinkle because they're closer, is at best half true. A tiny asteroid at the same distance as a planet would still twinkle. It's the apparent angular diameter that matters, and this gets larger when a given-sized object is closer, and also when an object at a given distance is larger.
@ Max: I too was mystified that an "avid amateur astronomer" could be so confused about twinkling. And just for the record, the blinking of (incandescent, and, I think, also fluorescent) light bulbs happens twice per cycle, not once, because the magnitude of the current reaches a maximum value at both (+ & -) peaks, and the bulb doesn't care which way the current is flowing. So In North America, where the line frequency is 60 Hz, bulbs blink at 120 Hz, and in Europe, where the line frequency is 50 Hz, they blink at 100 Hz. And obviously, regardless of their wattage.