The first way involves what is known as the Drake equation (named after radio astronomer Frake Drake). We start with the fact that there are about 200 billion (that's 200,000,000,000) stars in the Milky Way. (That's quite alot.) But not all stars are likely to be comfortable sites for life. About half are binary stars, and a planet in near a binary star does not have a stable orbit. Over time, the planet would be ejected out of the system into space, or thrown into a very eccentric orbit. Thus, life is unlikely to evolve around these systems.
Other types of stars also have problems. Any planet around a red giant star is likely to be gone, swallowed up by the star. Similarly, planets around white dwarf stars are unlikely (since these stars used to be red giants. Since terrestrial planets are made of heavy elements, one would suspect that planets around extreme Population II stars would be rare. Finally, main-sequence O, B, and A stars have very short lifetimes. For the first billion years of the solar system, planets were uninhabitable due to the constant bombardment of asteroids and comets. If that is typical then, by the time a proto-stellar disk is cleared of debris, upper main sequence stars would no longer exist. That leaves stars on the lower main sequence, especially F, G, and K stars. (I'm excluding M main-sequence stars for reasons I'll present below.) This excludes about 90% of all stars from consideration.
(Note that we now know that planets are common about such stars. Before 1994, astronomers knew of no extra-solar planets, but the observations of the last few have shown that many stars have planetary systems. It is true that the planets we are finding are Jupiter-sized objects, and that these objects most likely destroyed terrestrial planets during their migration into the inner solar system. However, it is possible that these object have moons that are habitable.)
Now let's consider the number of planets around a typical F, G, or K stars that are suitable and may develop life. What do we mean by "suitable for life"? Only bodies a certain distance away from their star (or another heat generating object) can develop life. Inner planets, like Mercury and Venus in the Solar System, are two hot for comfort; the ultraviolet light from the star would destroy most complex molecules. On the other hand, planets that are formed far from their star would be very cold: life requires energy, and, as one gets further and further from a star, there is less and less available energy. In other words, there is a habitable zone around all stars where life can develop. Around bright stars, this habitable zone is fairly large; around faint stars, it is small. In fact, around M main sequence stars, it is so small that it is virtually non-existant: any planet close enough to be warmed by the star, would also be in danger of being ripped apart by the star's tides. (Or, at ther very least, the planet would be tidally locked, keeping the same side of the planet towards the star at all times. This would result in a frozen atmosphere on one side of the planet, and an evaporating atmosphere on the other.) In addition, of course, a planet must have the appropriate mass to support life. Mars borders on the Sun's habitable zone, but the planet has too little mass to hold an atmosphere. (There is also a theory that says that habitable planets must have sizeable Moons, to stabilize their precession. Otherwise, over time, the warm cozy, equator of the planet might become the north pole!)
There are many details about planetary formation that are unknown at the present time. As a result, it is difficult to quantify the fraction of planets suitable for life. A rough guess might be that 1 in 10 planets might be suitable for life.
Next, we have to ask whether intelligent life will develop on the planet. Again, the conditions that give rise to the development of intelligence are unknown. One can guess that 1 in 10 planets with life develop intelligent life, but again, this is a matter of guesswork, and your guess is as good as mine.
Now consider a planet with intelligent life. Will this intelligent life develop a technologically advanced civilization, and want to communicate with the rest of the universe? Perhaps it will just live in the ocean and splash about having a good time. Perhaps it will decide that technology is bad. Perhaps 1 in 10 will want to communicate; perhaps more, perhaps less.
If you multiply these numbers together, the result is that there may be 20,000,000 stars in the Galaxy that have given rise to intelligent, communicating civilization. But, there's one more key factor: for what fraction of a star's lifetime will the civilization exist? Our Sun is about 5 billion years old, and for all but the last 50 years or so, we were not a communicating civilization. If we live for the rest of the Sun's lifetime (another 5 billion years), then we will have been an advanced, communicating civilization for 1/2 of our star's lifetime. On the other hand, if we destroy ourselves in the next 50 years, then the lifetime of our civilization would be only 1/1000000 of the Sun's life. If you believe that a typical civilization will last 1/2 its star's lifetime, then our Galaxy probably contains 10 million advanced communicating civilizations. In that case, the average distance between civilizations will be less than 50 light years. Our radio and TV signals may have already been picked up by our neighbors! On the other hand, if a typical intelligent communicating civilization lives only for a few hundred years, then we may be the only advanced civilization in the entire Galaxy.
You may wish to plug your own numbers into the Drake equation. You can do this, and see the equation written out explicitly by clicking here.
If there is a communicating civilization somewhere in the Milky Way, it is possible that we can pick up their radio waves. But what wavelength (or frequency) are they communicating at? How can we tell their signal from random noise? This is a difficult question, and requires a lot of computer computations. (In fact, this is a major limiting factor in the search for such signals.) One interesting method that the people who search for extraterrestrial intelligence has come up with is to develop a screen saver for people's home computer which would (in the background), analyze radio data and search for signs of intelligent life. If you would like to download this software, go to http://setiathome.ssl.berkeley.edu/download.html.
There is another, less traditional way of constraining the number of intelligent civilization in the Galaxy.
Think of the technological status of the earth 1000 years ago. Now extrapolate and consider of the technological status of earth 1000 years in the future, and all that might be possible. Now try to extrapolate 10,000 years, 100,000 years, 1,000,000 years, etc.) Keep that thought in mind, along with the fact that 1000 years is less than 1/10,000,000 the age of the universe.
Now, suppose there is at least one civilization out there in the Galaxy that is more advanced than earth. Suppose that just one of the creatures in one of these civilizations is rich and has the ego of a politician. This creature would like everyone in the Galaxy to know its place in history, i.e., it wants galactic fame. (Alternatively, the creature might want to start a new mega-advertising campaign for E.T. Cola -- the best selling Cola in the Universe.) So, now ask yourself whether this creature could:
The point of all this? If there were just 1 rich dude with an ego in just 1 advanced civilization out there, he could have easily populated the entire Galaxy with signs by now. And, if there were alot of rich dudes out there in alot of star systems, our solar system whould be cluttered with large, unmistable, signs. So, where are they? Unless there's a galactic ordinance against posting billboards in 5-billion year old solar systems, and/or a galactic police force charged with painting over the graffitti left by other civilizations, there should be very clear, unmistakable evidence of ETs in plain view. Since we don't see this evidence, one might almost conclude that there are no ETs with egos. (Or, the concept of an advertising campaign is unique to earth.)