Stellar parallax is the apparent shift in a star's position against background stars as Earth orbits the Sun over six months. Distance in parsecs equals one divided by the parallax angle in arcseconds. One parsec is approximately 3.26 light-years. Parallax is the first rung of the cosmic distance ladder, reliable to a few thousand parsecs; beyond that, other methods such as Cepheid variables and standard candles must be used.
Work through the geometry: draw Earth's orbit, a nearby star, and distant background stars, then calculate distances for given parallax angles. Compare parallax measurements with known stellar distances to appreciate the precision limits of ground-based versus space-based astrometry.
You already understand right-triangle trigonometry and how to calculate distances using angles and known baselines. Stellar parallax applies exactly this logic to measure cosmic distances, using the largest baseline available to us without leaving the solar system: the diameter of Earth's orbit around the Sun.
Here is the geometry. Observe a nearby star in January, then observe it again in July when Earth has moved to the opposite side of its orbit — a baseline of 2 AU (about 300 million km). The star appears to shift slightly against the backdrop of much more distant stars, which are so far away they seem fixed. This apparent shift is the parallax. The parallax angle (p) is defined as *half* the total angular shift, corresponding to a right triangle with a baseline of 1 AU (Earth-to-Sun distance) and the star at the far vertex. From basic trigonometry, if the angle is small (and it always is — we are talking fractions of an arcsecond), the distance d ≈ 1/p, where d is in parsecs and p is in arcseconds. One parsec — the distance at which a star would have a parallax angle of exactly one arcsecond — equals about 3.26 light-years.
The inverse relationship is the critical intuition: closer stars show larger parallax shifts, farther stars show smaller ones. The nearest star system, Alpha Centauri, has a parallax of about 0.75 arcseconds, giving a distance of 1.33 parsecs. A star with a parallax of 0.1 arcseconds is 10 parsecs away. A star at 0.01 arcseconds is 100 parsecs away. As stars get more distant, the angular shift shrinks until it becomes indistinguishable from measurement noise. Ground-based telescopes are limited to roughly 100 parsecs because atmospheric turbulence blurs stellar positions. The Hipparcos satellite extended reliable parallax to about 1,000 parsecs, and the Gaia mission has pushed precision to tens of thousands of parsecs for the brightest stars.
Parallax is the first rung of the cosmic distance ladder — the foundation on which all other astronomical distance measurements rest. Cepheid variable stars, Type Ia supernovae, and other "standard candles" are calibrated by first measuring their distances with parallax, then using their known luminosity to estimate distances where parallax fails. If the parallax rung is wrong, every subsequent rung inherits the error. This is why space-based parallax missions like Gaia are so important: by measuring billions of stellar parallaxes to microarcsecond precision, they tighten the calibration of the entire distance scale and, by extension, our understanding of the size and expansion rate of the universe.