The Masses of the Sun and Stars

The most reliable way in which to determine the masses of celestial objects is to study objects which are in binary systems. For example, the mass of the Sun can be determined by studying the motion of the Earth about the Sun; the Earth does not fall into the Sun because of its orbital motion. The gravitational pull of the Sun is counteracted by the centrifugal force generated by the orbital motion of the Earth. The stronger the pull of gravity, the faster the Earth must move in order to keep from falling into the Sun. This is illustrated by the figure to the right (modeled after one found in Newton's Principia). Imagine a cannon fired from a mountaintop. As the cannonball moves around the Earth, the Earth pulls it downward. However, as it falls downward, the ground, simultaneously, falls away from it. If the cannonball falls at the same rate as the ground falls away, then the cannonball never hits the ground and the cannonball orbits the Earth.
See an illustration of this using the applet on the UCLA astronomy site Newton's cannon.
To get a feel for how this works, play with the planetary orbital simulator provided by the University of Colorado, Phet simulator

Because the pull of the Sun's gravity depends upon its mass and how far we are away from the Sun (the A.U.), we can use the speed of the Earth to infer the mass of the Sun. Formally, we use Newton's formulation of Kepler's Third Law of Planetary Motion, i.e.,

P² = 4π²a³ / [G (M_Sun + M_Earth)]

where P is the orbital period of the planet, π = 3.14, a is the semi-major axis of the planet's orbit and G is the gravitational constant. The semi-major axis a is the average of the closest and fartheset approach of the stars about the center-of-mass of the system, where the center-of-mass is defined below,
. The value for G depends upon the unit system one chooses. The mass of the Sun is

M_Sun = 2 x 10³⁰ kilograms

This method is also used to determine the masses of nearby binary stars. To get a feel for how this works, use the binary star orbital simulator written by Michael Topping (Binary Star Simulator written by Michael Topping on UCLA Astronomy, https://www.astro.ucla.edu/undergrad/astro3/orbits).

When

one of the objects is much less massive than the other (as in a star/planet or planet/moon situation), we only see the low mass object move which tells us the gravitational pull of the more massive object and so tells us its mass. In this case, we learn the mass of the star or the planet.

the objects are of comparable masses, we can see the motion of both objects moving in the gravitational pull of the other object and so we can learn the masses of both objects!