1. Using RA and dec
All stars and other objects in the sky are at different distances from earth. But if we for a moment forgot that and instead imagined them on the inside of a great sphere surrounding the earth then we'd have something called a celestial sphere. Now what if we wanted to describe the positions of those stars and objects? Since the celestial sphere surrounds the earth, let's do a few things. First, extend the earth's equator out until it intersects the celestial sphere. Now we have the celestial equator. Do the same thing with the earth's poles and we have the celestial poles. On earth, we can measure latitude as how far north or south something is from the equator. If you are at the equator you are at 0° latitude. At the North Pole, you are at +90°. So latitude is measured from 0 to ±90°. We can do the same thing on the celestial sphere, but instead we call it declination. Declination is also measured from 0° at the celestial equator to ±90° at the celestial poles.
A little tougher to translate is longitude. On earth we define it as how far east or west a place is from the Prime Meridian which is defined as a line beginning at the North pole, passing through Greenwich, England (UK), then ending at the South pole. There are historical reasons for that. But what about in the sky? Is there a 'Prime Meridian' up there? Yes, there is but first we need to define some other points in the sky.
Visualize the earth with the celestial sphere surrounding it and the celestial equator and poles are marked. Now, I bet you're imagining this with the earth's axis perfectly vertical. Remember, the earth's axis is tilted in relation to its orbit. So now it's tougher to imagine, but running across our celestial sphere is another line called the ecliptic. One way of 'seeing' this line is to imagine the earth and celestial sphere going around the sun as one unit. As the earth moves around the sun, the sun is moving across the celestial sphere. When the earth is tilted away from the sun (northern hemisphere winter), then the sun and ecliptic are below the celestial equator. When the earth is tilted towards the sun (northern hemisphere summer), the sun and ecliptic are above the celestial equator. And then there are those two times when the axis is not tilted towards or away from the sun (spring and fall) so it looks like the sun and ecliptic are crossing the celestial equator. The winter and summer points are called the solstices and the spring and fall points are called the equinoxes. Because springtime is a time of birth and renewal, the Spring or Vernal equinox was considered to be very important so it was natural that when astronomers were trying to figure out a 'Prime Meridian' for the sky that they would choose a meridian passing through the Vernal Equinox. Because the vernal equinox is the point in the sky where the ecliptic crosses the celestial equator from south to north or is ascending, we call these longitude-like lines 'lines of RA or right ascension.'
So now we know where the 0 point or prime meridian is in the sky. How do we measure RA in the sky? Remember, on earth we measure longitude east and west from the Prime Meridian. In the sky, we measure it slightly differently. Earth is slowly turning on its axis -- once every 24 hours. Let's say that one day the Vernal equinox is crossing our local meridian (the line running north-south through the zenith, separates the sky into east and west). An hour later, we can imagine that crossing our meridian is another line of RA. And every hour after that, we have another line of RA overhead until we're back where we started. So we number these lines of right ascension in hours from 0 to 24.
So how do we use RA and dec for finding an asteroid or comet? Well, sometimes, you might not be given a chart that shows your target's position. Instead you might be handed a list of numbers, an ephemeris, which includes the target's RA and dec maybe every 30 minutes during the night. You can use the RA and dec to plot the position of the asteroid or comet on your charts so you know where to look for it. Or, if you have a computerized telescope you can enter the coordinates into the computer and let the telescope slew to the target. This is a very lazy way of doing things and does not insure that you will be looking at your target. If you are familiar with charts and star-hopping, then you'll be able to verify that you are looking at the asteroid or comet! This is particularly important when searching for very dim objects that might not be visible to the eyes even in the telescope, but become visible in photographs through the telescope.