(Abbreviated as RA, or occasionally as lower-case Greek letter Alpha: a)
Right Ascension is a coordinate on the celestial sphere that is similar to, but not identical to, longitude on the Earth's surface. Right ascension measures the positions of celestial objects in an east-west direction, like longitude, but unlike longitude right ascension is a time-based coordinate.
The other celestial coordinate, Declination, is similar to the terrestrial coordinate latitude. A diagram of Right Ascension and Declination appears below:
As the Earth rotates on its axis, the celestial sphere appears to revolve around the Earth, making one complete revolution in one sidereal day (23 hours, 56 minutes, 4 seconds). A sidereal day is thus about 4 minutes shorter than a mean solar day. This time difference between a sidereal and a solar day is the result of the Earth moving 1/365th of the way around the sun during this period.
Think of the celestial sphere as a giant plastic ball with the Earth at the center. The stars are painted on the inside of the plastic ball, along with the lines of the celestial coordinates Right Ascension and Declination. The ball does not move as the Earth turns in the center, but as we here on Earth see it, it looks like the ball (the celestial sphere) is turning around the Earth. Because the ball is just sitting there, the things that are painted onto the ball (stars and coordinate lines) do not move in relation to each other. The whole celestial sphere, stars, coordinate lines, and everything, appear to us on Earth to move together, making a complete circle every sidereal day.
Right Ascension is essentially a time measurement. You can think of RA in this way: Whenever the point on the celestial sphere that we have set as the "start" of Right Ascension transits our local meridian, start a stopwatch. When the celestial object of interest (a star, for example) transits our local meridian, stop the stopwatch. The time on the stopwatch is that object's Right Ascension. Right Ascension is expressed in units of time on a 24 hour format. A star could have a RA of 17h 32m , for example. This would mean that the star transited our meridian 17 hours and 32 minutes after the "start" of Right Ascension transited.
Where IS the "start" of Right Ascension? Astronomers needed to pick someplace on the celestial sphere to start timing for Right Ascension. The most obvious place is one of the points on the celestial sphere where the two principle celestial paths, the ecliptic and the celestial equator, cross. There are two such points, one when the ecliptic moves from south of the celestial equator to north of the celestial equator (known as the point of the Vernal Equinox) , and one where the ecliptic moves from north of the celestial equator to south of the celestial equator (known as the point of the Autumnal Equinox.)
The point chosen was the point of the Vernal Equinox. It is important to understand that the term "Vernal Equinox" can refer to two different things. In this situation we mean the point on the celestial sphere where the paths of the ecliptic and the celestial equator cross near the constellation Aries. (The term "Vernal Equinox" can also mean the moment in time when the sun is actually located at that point, but this is not the meaning in this context.)
Astronomers use Sidereal Time to measure the movement of the celestial sphere. Local sidereal time, that is the sidereal time where we are located, is equal to the right ascension of any celestial object that is transiting our meridian at this particular moment.
Avoid a common misunderstanding:
Celestial objects (stars, planets, etc...) have a Right Ascension. They do NOT have a sidereal time. It does not make sense to say, for example, that the Moon's sidereal time is 15h 00m. The Moon does not have a sidereal time. The Moon (or the sun, stars, etc.. does have a Right Ascension.
Locations on the Earth have a Sidereal Time. They do NOT have a Right Ascension. It also does not make sense to say, for example, that the Right Ascension of Raleigh is now 21h 00m. Places do not have a Right Ascension.
Look at the diagram below. Let's say that star 2 has a Right Ascension of 06h 00m. The entire celestial sphere slowly moves from left to right over the course of the night, so the positions in relation to the horizon and the meridian of all the stars shown slowly change.
In the diagram, star 2 is just now transiting our local meridian, so our local sidereal time is 06h 00m.
Star 1 has a right ascension of 05h 00m. Star 1 already transited 1 hour ago. When star 1 was on our meridian, our sidereal time then was 05h 00m.
Star 3 has a right ascension of 07h 00m. When star 3 transits our local sidereal time will be 07h 00m. Since our local sidereal time is now 06h 00m, star 3 will transit in one hour.
So you see that the celestial sphere is like a giant clock keeping sidereal time.