How do we know how far away a star is?
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There are several ways:
When measuring relatively close stars, we use parallax measurement. Take an accurate measurement of a stars position, and wait 6 months. Take that measurement again. Just like when you close one eye and then the other, the apparent location shifts because of the changing viewing angle. We can use the known diameter of the orbit of the earth to figure out the stars distance.
There are certain types of objects which are referred to as “standard candles”. These are objects or events which, thanks to their physical properties (which have been mathematically figured out), we know their intrinsic brightness, or absolute magnitude.
Type 1a supernova are caused when a binary system has one large star and one small. The large pulls matter off the small, and at a very predictable amount it all ignites in nuclear fusion. This always happens with the same amount of mass, so the brightness is always the same. By comparing that with the apparent brightness, we can find the distance.
Cepheid variable stars are stars which fade in and out, the time it takes is directly related to the maximum brightness. Time them and you can see how bright the absolute magnitude is, and again compare to the apparent brightness.
There are many other methods, using gravitational effects, spectroscopy, Doppler effects etc etc but these are the main ones.
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