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It started with Newton in 1687, who gave us the universal gravity equation :
F = G m1 m2 / r^2
In 1797-8 Henry Cavendish computed the value of G (universal Gravitational constant) :
http://en.wikipedia.org/wiki/Henry_Cavendish
Back in 1798 he "weighed" the earth.
see the above article for how he did it.
With that it was possible to compute the weight of any other solar system body ie any of the planets, the sun, all the moons, asteroids and comets.
His 1798 value for G is within 1% of the currently accepted value
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Our estimation is only as accurate as the gravitational constant can be measured in laboratories. This is not an easy thing to do, since gravity is such a feeble force. So we're only able to estimate it accuratly to a about 5 significant digits.
The sun's mass itself is determined by the orbital speeds of planets, just like the mass of planets are determined by the orbital speeds of their moons.
Harley Drive
like the other answer it's pure guesswork