Measuring the Sky
Astronomers need a system to describe positions in the sky so they can report to each other where to look for things they have found. The basic unit used in the celestial coordinate system to describe positions -- and the distances between them -- is the degree of arc, just like the degrees used to give latitude and longitude on the Earth's surface. One degree is divided into 60 minutes of arc, each of which is in turn divided into 60 seconds of arc. The equivalents of latitude and longitude in the sky are declination and right ascension, respectively. The second of arc is also the basis of another commonly used astronomical distance, the parsec. As Earth moves in its orbit, nearer stars appear to shift position relative to the more distant background stars. This is called parallax. A parsec is the distance from Earth that would shift exactly one parallax second of arc over the Earth's orbit.
The Speed of Light
The speed of light is a useful basis for astronomical distances, because light travels so fast (186,283 miles per second in the vacuum of space, a little slower through air) and because it is such an important constant in astrophysics for so many other reasons. A unit of time, multiplied by the speed of light, becomes a convenient unit for astronomical purposes. The Moon is about 1.3 light-seconds away, the Sun about 8 light-minutes, and Alpha Centauri, the nearest star system to our own, 4.3 light-years. There are about 3.26 light years to one parsec.
Familiar Equivalents
Most measurement systems use units based on things of a familiar size, the imperial foot being a prime example. Astronomers also use familiar equivalents to talk about various measurements in space. For distance, the Astronomical Unit (A.U.) is defined as the mean distance from Earth to the Sun, and is a convenient way to compare distances on an interplanetary scale. Similarly, the masses of Earth and the sun are useful equivalents when talking about the masses of other planets and stars. (Earth's mass is about 1.3 x 10^25 pounds, and the sun is approximately 333,060 Earth-masses.)
Other Units of Measurement
There are many other highly specialized units for the sorts of things astronomers are interested in measuring. The brightness of a star, for example, is measured in relative magnitude (how bright the star looks from Earth) where first-magnitude stars are about 2.5 times brighter than second-magnitude stars. There is also absolute magnitude, defined as how bright the star would look if it were 10 parsecs away. Cosmologists measuring the expansion of the universe have units derived from the Hubble constant, and deal in both the very large (billions of light-years) and the very small (nanoseconds after the Big Bang). There are measurements of wavelengths and frequencies, gravitational and magnetic fields, redshifts, temperature, pressure, density, and many more, most of which are based in the same units we use to measure the same properties on Earth.