The sun is unusual in being alone. Something over half of the stars in the sky come in pairs or larger groups, held together by their own gravity and going round each other for as long as they live.
It is tempting to picture the lighter star going round the heavier one, but that is not what happens. Both go round a single point that lies between them, the center of mass, and neither of them is still.
The sun and the earth do the same thing. The sun swings round the center of mass of the pair once a year. It happens that this point lies inside the sun, so the swing is small enough to overlook.
The closer the two masses are to each other, the closer the center of mass is to the midpoint. For two equal stars it sits exactly halfway.
The center of mass sits nearer the heavier star, and by a fixed amount. If a_1 and a_2 are the distances of the two stars from it, then M_1 a_1 = M_2 a_2.
So the heavier star traces the smaller circle. Watch the pair for one full turn, measure how far each one swings, and the ratio of the swings gives the ratio of the masses directly. Nothing else is needed.
But a ratio alone does not give either mass. Knowing that one star is 3 times the other leaves every pair from 3 and 1 up to 30 and 10 still open.
What is missing is the sum. Once the ratio and the sum are both known, each mass follows. The sum is what the next step supplies.
Kepler found that for the planets, the cube of the size of the orbit is proportional to the square of the period. Newton showed why, and showed what the constant of proportionality is: it is the total mass.
Measured in units where distances are astronomical units, times are years and masses are suns, the relation is exactly \dfrac{a^3}{P^2} = M_1 + M_2.
For a planet going round the sun, the total is essentially 1, so a^3 = P^2. The earth at a = 1 takes P = 1, and that is the whole of it.
For a pair of stars the total is not 1, and the difference shows up as a period that is shorter or longer than the planetary case would give. That difference is the total mass. With the ratio from the swings and the sum from here, both masses are fixed. This is the only direct way there is to weigh a star.
If the plane of the orbit happens to lie close to our line of sight, then once every turn each star passes in front of the other. The pair is called an eclipsing binary.
The light arriving drops while one is hidden, and the record of it over one turn is the light curve. There are two dips, one for each crossing, and they are not equally deep.
The deeper dip is the one where the hotter star goes behind. Both crossings hide the same area, but hiding the hotter surface takes away more light. Which dip is deeper says which star is hotter.
The width of a dip gives the size of the stars, since it is the time taken to cross the disc. So an eclipsing binary yields masses and radii together, and those pairs are what every model of stellar structure is checked against.
Most binaries are far too close together to be seen as two. In a telescope they are a single point of light, and nothing about the image says there are two stars there.
The spectrum says it. As one star swings toward us its absorption lines shift toward the blue, and as it swings away they shift toward the red. This is the Doppler effect, and the shift is proportional to the speed along the line of sight.
Because the two move in opposite directions at every moment, their two sets of lines pull apart and close again once per turn. The ratio of the two shifts is the ratio of the two speeds, which is again the inverse ratio of the masses.
One thing is still missing. A tilted orbit shows only the part of the motion along the line of sight, so what comes out carries a factor of \sin i, where i is the tilt. If the pair also eclipses, the tilt is known to be near edge on, and then the masses come out clean. Those are the systems the whole mass scale rests on.