How is the mass and weight of objects in Space measured-while we're on the subject, how much does the Sun weigh?

There I moved your question to the right category.

Mass of objects in space is estimated by using an algorithm that considers the size of the object, the material it is mainly made of, and the compression of the object's own gravity. The mass of the Sun is 1.989 × 10^30 kg, more-or-less.

Objects only have "weight" when they are in a gravitational field. Think of it like this: An object's weight is the force it would apply on the platform of a scale, with the scale on some planet's surface (so subject to that planet's gravity). If you had a scale so large (and so heat-proof) that you could put the Sun on it, there would be no gravity, out where the Sun is, to pull the Sun and cause it to press down on the scale's platform. So the Sun would "weigh" nothing, despite its enormous mass.

On Earth we only have to weigh the object and divide by the gravitational acceleration, but this obviously doesn't work in space. To measure mass in space, we have to use another kind of scale, which is called an inertial balance. An inertial balance is made of a spring on which you attach the object whose mass you're interested in. The object is therefore free to vibrate, and for a given stiffness of the spring the frequency of the vibrations enables the scientists to calculate the mass.

This is how you would get the mass of objects in a space shuttle, or something like it. But there are other objects in space that astronomers are very interested in knowing their masses: stars and galaxies. The way to get the mass of these objects is to look at the gravitational interaction with other objects nearby. For example, if you have two stars orbiting one another and you know the distance between them and how long it takes for one to go around the other, you can calculate the mass of the stars. Similar tricks apply to measure the mass of galaxies, for example by measuring how fast they rotate.

And to answer your second question: the sun weighs 1.989 × 10^30 kg.