the sun - What is the exact mass of the Sun?

The mass of the Sun is determined from Kepler's laws:

$$frac{4pi^2times(1,mathrm{AU})^3}{Gtimes(1,mathrm{year})^2}$$

Each term in this component contributes to both the value of the solar mass and our uncertainty. First, we know to very good precision that the (sidereal) year is 365.256363004 days. We have also defined the astronomical unit (AU) to be 149597870700 m. Strictly speaking, the semi-major axis of the Earth's orbit is slightly different, but by very little in the grand scheme of things (see below).

At this point, we can solve for the product $GM$, known as the gravitational parameter, sometimes denoted $mu$. For the Sun,
$$mu_odot=132712440018pm9,mathrm{km}^3cdotmathrm{s}^{-2}$$

So solve for $M_odot$, we need the gravitational constant $G$, which, as it turns out, is by far the largest contributor to the uncertainty in the solar mass. The current CODATA value is $6.67384pm0.00080times10^{-11},mathrm{Ncdot m}^2cdotmathrm{kg}^{-2}$, which all combines to give
$$M_odot=1.98855pm0.00024times10^{30},mathrm{kg}$$
where my uncertainty is purely from the gravitational constant.

The value $1.9891times10^{30},mathrm{kg}$ (and nearby values) probably come from an older value of the gravitational constant of $6.672times10^{-11},mathrm{Ncdot m}^2cdotmathrm{kg}^{-2}$, which is still supported by some measurements of $G$.