radial-velocity method
How do you find a planet you cannot see, around a star a trillion times brighter? You watch the star wobble. A planet does not orbit a perfectly fixed star; the two circle their shared balance point, like a hammer-thrower and the hammer both swinging around a center between them. The star, being far heavier, makes only a tiny loop — but it does move, toward us and then away, again and again. The radial-velocity method detects a planet by measuring that small back-and-forth motion of the star along our line of sight.
The trick is the Doppler shift: when the star edges toward us its light is squeezed slightly bluer, and when it edges away its light is stretched slightly redder, just as a passing siren rises then falls in pitch. By splitting the star's light into a spectrum and watching its sharp dark absorption lines slide back and forth, astronomers measure the star's speed toward and away from us to astonishing precision — better than walking pace, around 1 meter per second, on a star light-years away. The size of the wobble reveals the planet's mass (strictly, a minimum mass, because we usually don't know the tilt of the orbit), and the rhythm reveals its orbital period.
This was the method that found the first planet around a Sun-like star, 51 Pegasi b, in 1995, and it remains a workhorse, especially good at weighing planets and finding worlds that do not happen to pass in front of their stars. Its great limitation is that it gives a minimum mass, not a true one, unless the orbit's tilt is known from another method — and it favors massive, close-in planets, which yank their stars hardest, while small or distant worlds make wobbles almost too faint to detect.
Jupiter makes the Sun wobble at about 12 meters per second; the Earth makes it wobble at only about 0.09 meters per second — slower than an ant walks. Detecting an Earth-twin by radial velocity means measuring a star's speed to better than the crawl of an insect, which is why such worlds remain at the edge of what the method can do.
The method reads a planet's pull from the tiny Doppler wobble it gives its star.
Radial velocity gives a minimum mass, not the true mass, unless the orbital tilt is known — and it cannot tell you a planet's size, only how hard it pulls.