From a number of studies, the mass of the Milky Way can be estimated to an accuracy of perhaps 20 percent as 140 billion times the mass of the Sun, if you use the Sun's speed around the core of the galaxy. Radio astronomers have detected much more material outside the orbit of the Sun, so the above number is probably an underestimate by a factor of 2 to 5 times in mass alone.
Now, to find out how many stars this represents, you have to divide by the average mass of a star. If you like the sun, then use 'one solar mass' and you then get about 140 billion sun-like stars for what's inside the sun's orbit. But astronomers have known for a long time that stars like the sun in mass are not that common. Far more plentiful are stars with half the mass of the sun, and even one tenth the mass of the sun. The problem is that we don't know exactly how much of the Milky Way is in the form of these low-mass stars. In text books, you will therefore get answers that range anywhere between a few hundred billion and as high as a trillion stars depending on what the author used as a typical mass for the most abundant type of star. This is a pretty embarrasing uncertainty, but then again, why would you need to know this number exactly?
The best estimates come from looking at the motions of nearby galaxies such as a recent study by G. R. Bell (Harvey Mudd/USNO Flagstaff), S. E. Levine (USNO Flagstaff):
Using radial velocities and the recently determined proper motions for the Magellanic Clouds and the dwarf spheroidal galaxies in Sculptor and Ursa Minor, we have modeled the satellite galaxies' orbits around the Milky Way. Assuming the orbits of the dwarf spheroidals are bound, have apogalacticon less than 300 kpc, and are of low eccentricity, then the minimum mass of our galaxy contained within a radius of 100 kpc is 590 billion solar masses, and the most likely mass is 700 billion. These mass estimates and the orbit models were used to place limits on the possible maximum tangential velocities and proper motions of the other known dwarf spheroidal galaxies and to assess the likelihood of membership of the dwarf galaxies in various streams.
Again, you have to divide this by the average mass of a star...say 0.3 solar masses, to get an estimate for the number of stars which is well into the trillions!
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