The massive black hole
By following the motions of individual stars we have measured the mass associated with radio source Sgr A* - we see stars on Keplerian ellipses surrounding an object of 4 million times the mass of the Sun. The physics at play is extremely simple: It is Newton's law of gravity. The precision of these measurements is stunning - see the orbit of the star S2 (Gillessen et al. 2009, Gillessen et al. 2013)
S2 is the best example of the orbits, and constraints the mass most. Overall, we can measure the mass with a precision of around 1%. Furthermore, we can locate the mass and show that its position agrees to better than 1mas with where the radio source Sgr A* is located (Plewa et al. 2015). This measurement uses a few SiO maser stars, which are visible both in the infrared, and at radio wavelengths.
Since we measure proper motions (angle on sky per time) and radial velocity (km/s) for the stars, we actually can constrain geometrically the distance R0 to the Galactic Center. The following figure shows that we reach a precision of few percent (Gillessen et al. 2013).