To find where the nearest stars are, you use parallax. If you look at a nearby object with one eye open, and suddenly switch eyes, you'll notice that the object seems to move. This is parallax: and if you know how much the object moves, and you know how far apart your eyes are, you can figure out how far away the object is. (It's basically a geometry problem).

Astronomers will take a picture of a star, wait 6 months, and compare the position to the previous one. The star will shift a little because the Earth has moved about 300 million kilometers in the intervening time. Knowing how far the Earth moves in half a year, and measuring the angular shift of the star, you can calculate its distance. This method only works for stars closer than about 300 lightyears. Other methods are used to check stars that are further away.

Question: If you were doing this technique on Mars, would it get better or worse results?

There are various ways to measure distance.For very close objects we can use radar in which we shoot radiation at something and time how long it takes to reflect off an object and get back. Since we know the speed of light we can measure distance (If it takes one second to get our signal back and the speed of light is 186,000 miles per second how far away is the object?)

For farther off objects we use something called parallax. To see how this works, go outside and look at an object that is relatively close by, like a tree, and some far off object behind it such as a mountain or building. Remember what part of the far off object the tree covers and then move about ten feet to your right or left. Does the tree seem to move with respect to the farther away object? You can measure the distance to a star by measuring how much it seems to move when looking at it from two different places on Earth at the same time just like you could figure out the distance to the tree by seeing how much it seems to move.

For objects so distant that the parallax is too small to measure we use objects called standard candles. Imagine that you are looking a street light that is close enough to you that you could measure the parallax. Then imagine that you see another street light a longer distance away. Since the far away street light should be just like the close up one, you should be able to figure out how far away the farther one is based on how much dimmer it is. This is basically what is done to measure the distance to really far away objects. The hard part is figuring out what you can use as a standard candle.

This is a question upon which a lot of research is done. There are a few techniques. The first is called parallax. Hold your arm out in front of you and hold up one finger. Now keep one eye opened and close the other. Now switch which eye you have opened while looking at what is behind your finger. You will notice that your finger appears to move with respect to the stuff behind it. From the amount of change you can actually figure out how far away your hand is!! Try drawing a picture of what is going on. (Hint: you need 5 points and 2 lines, 1 point for each of your eyes, one for your finger, and one for each thing that you saw behind your finger. then just connect the right dots)

Other techniques use the fact that things that are far away from us are moving away from us because the Universe is expanding. If you know how bright something is and how fast it is moving you can know how far away it is.