I’ve made a powerpoint presentation here that demonstrates the astronomical principle of trigonometric parallax.
For the uninitiated, parallax is a fancy term for triangulation, which we humans naturally do in the form of depth perception – your eyes are separated by a small distance from each other so they see slightly different views of the world. You can try it now by closing one eye, holding out your thumb, and lining it up with a mark on a far wall, then switching to the other eye. Your thumb is now no longer lined up with what it was originally; that shift is called parallax. (Also: consider ATMs, and how the on-screen captions NEVER seem to be lined up with the buttons.)
Where parallax plays into astronomy is that it’s a great way of measuring really large distances, like the distances to stars. To do this, we actually use the Earth’s orbit as a baseline – observing a star in January, and then again in July when the Earth is on the other side of the Sun (as seen in the animation). We see the star from vantage points 300 million kilometers (184 million miles) apart, and the star’s position on the sky appears to change very slightly.
Astronomers as early as the ancient Greeks looked for and expected to find this motion (if the stars were hanging at non-infinite distances they SHOULD wobble back and forth – this was originally one of the problems with the heliocentric model of the Universe). Unfortunately, stars are so incredibly far away that the parallax motions are measured in increments smaller than an arcsecond – 1/60 of 1/60 of 1 degree of the sky (the moon is 1800 arcseconds across; the human eye can see details as fine as 1/60 of a degree). It took until the 1830s (more than 200 years of innovation and improvements in technique) for astronomers to accurately measure angular displacements that small. Today, telescopes routinely measure positions accurate to thousandths and millionths of an arcsecond, enabling us to determine accurate distances to objects thousands of light years away.