Their eyes,
Refracting Telescopes (ones with class lenses)
Reflecting Telescopes (ones with mirrors)
Radio Telescopes
Imaging computer chips in conjunction with telescopes
Space based telescopes
Underground telescopes (to detect high energy or exotic particles from stars)
Gravity wave detectors
Astronomers use apparent magnitide to measure the brightness and dimness of a star, first magnitude stars are the brightest, and can be visible as soon as the sun goes down, but sixth magnitude stats do not appear to be visible until some time after sunset, and are always invisible on moonlit nights and in light polluted cities.
the astronomers use Absolute magnitude
Astronomers use a method called parallax to measure the distance to nearby stars. Astronomers can measure parallax by measuring the position of a nearby star with respect to the distant stars behind it. Then, they measure the same stars again six months later when the Earth is on the opposite side of its orbit.
They use the banana method
I have usually seen them quoted in kilometers.
Distance.
Because the angle of shift is very small and extremely difficult to measure, scientists usually use photography to measure the shift. The star is photographed at the beginning and end of a six-month period, and its position in relation to other stars is studied each time.
Because they cannot visit stars and measure their sizes with a tape measure!
the sun
Astronomers use a method called parallax to measure the distance to nearby stars. Astronomers can measure parallax by measuring the position of a nearby star with respect to the distant stars behind it. Then, they measure the same stars again six months later when the Earth is on the opposite side of its orbit.
They use the banana method
Usually light-years, or parsecs.
I have usually seen them quoted in kilometers.
They do not use units: they use the fact that stars are not galaxies. For example, you don't use units to measure the difference between children and countries.
Distance.
The device that astronomers use to find the angle between the horizon and stars in the sky is called a sextant.
With a long measuring tape.
the astronomers use Absolute magnitude
Astronomers use the unit of measure "Light Years" to calculate the distance between pretty much anything in the Universe.