This is a good question, and something that people are still working on. This is in general a complicated question, but I will try to give you a couple of ideas about where to start.
One method is the method of parallax; if you hold your finger in front of your face, and move your head back and forth, the apparent motion of the scene behind your finger relative to your finger tells you something about the relative distances involved. If you know how far you moved your head, you know even more.
Another method uses the doppler principle and Hubble's law. Stars that are moving away from us end up with their colors being shifted towards the red end of the spectrum. Stars moving towards us have their colors shifted towards the blue end of the spectrum by the Doppler effect. Hubble's law, which has been verified a few ways, is that the faster a star is moving away from us, the further away it is.
Another thing that goes into these estimates are the laws of gravity and motion, like Kepler's laws. We assume that the objects we are looking at have to obey these physical laws (or sometimes more accurate ones like those stemming from general relativity).
Of course, sometimes mistakes are made and later information forces scientists to revise their estimates. But hopefully after a few years, the estimates about distances in space become more and more accurate.