Triangulate the distance with geometry. If you know the distance between two points in a triangle and the angle between those two and the third, you can calculate the distance between all three.
We know the distance to the sun (the nearest star) because we dropped mirrors all over the moon and used them to bounce signals off and then time how long they take to go there and back, calculate the distance and then we've got two sides of the triangle and can figure out the third.
We can do the same with other stars.
The method outlined above is called "parallax". It is how we establish distances to stars near earth. Since the earth is moving around the sun (and as the guy above says we know the distance to the sun) we can use the "width" of the earths orbit as what we call the baseline. (the bottom of the triangle.) near earth stars (near still being very far away, the order of parsecs) will appear to move against the background of distant stars. We can take measurements from each "end" or the orbit and use them to calculate (as above) the distance to the star.
This is best demonstrated by holding your finger up in front of your face and looking at it through one eye at a time, the distance between your eyes serving as the baseline. if you could measure the apparent movement of your finger and you knew the distance between your eyes you could work out how far your finger was from your face.
Once we have the distance to a few nearby stars like this we start to use more outlandish methods.
Stars tend to fit into various types depending on colour (which indicates temperature) and age. If we know the distance to a star (by parallax) and we know what type it is we can use what is called the "standard candle" method to work out the distances to further away stars of the same type.
So - I know that a "type G2V" (the sun) is X bright if it's Y far away... i assume that a G2V which appears much dimmer is much further away - we know that brightness decreases related to the square of distance (can't remember the full equation of the top of my head) so you can use this method to pin down distances for loads of stars provided you can get a fixed reference the "standard candle" - obviously this method has issues, mostly to do with the light from stars being dimmed for reasons other than distance like dust in the way. However it's the best we've got!
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