Two basic principles guide the Search for Extra-terrestrial Intelligence (SETI) and the debate about whether we are truly alone in the universe. The first, the Drake equation, is generally an optimistic calculation that there should be numerous alien civilizations alongside ours, elsewhere in the galaxy. However, the second principle, the Fermi Paradox, tends to throw cold water on that idealism. If intelligent aliens are out there, then it should be relatively easy to find evidence of them, states the paradox. However, we have not found such evidence (or, at least, we have not been able to confirm any alleged evidence). Therefore, either we were wrong to assume that life is plentiful, or we somehow lack the tools to find it.
The name for the paradox allegedly comes from a conversation at Los Alamos involving the famous physicist Enrico Fermi. Given the scale of the galaxy (hundreds of billions of stars), it was foolish to believe that there were not at least some other stars with conditions similar to our own, and which had given rise to intelligent species like humanity, agreed Fermi's colleagues. However, at some point Fermi interjected a question: if that was true, then were they?
Fermi's point was a logical one. Given that the galaxy is billions of years old, if there are numerous alien civilizations, then at least some of them should be significantly older than humanity. Moreover, if the increasingly rapid technological progress which we have seen in recent centuries is going to continue into the future, then it stands to reason that a civilization even just a few centuries or millennia older than us could have technology exponentially more powerful than our own. (Consider: a century ago we were experimenting with powered flight into the air, whereas now flight into orbit is technically routine, if still complicated and very expensive.) Such a civilization should therefore have spread widely around the galaxy. Even if we had not made contact with an alien explorer or diplomat, we should have had no trouble spotting telltale evidence of their presence in the galaxy.
Yet, so far, we have not - and, fifty years after Fermi's debate, we still have no such evidence. This suggests that something has gone wrong in our thinking. Either we were mistaken about how common life is, or we were mistaken about our ability to detect other life forms.
The simplest explanation, and therefore the easiest to accept at present, is simply that other intelligent races do not exist, or at least that they are incredibly rare and so far apart that we may never see each other at all - or at least, not without many more decades or even centuries of deliberate searching. Perhaps the confluence of fortunate events in our solar system, like the protection given by Jupiter, or the many adaptations which eventually led to the emergence of human beings, is extraordinarily rare. This is known as the "Rare Earth" theory or, in some forms, as the "Great Filter" theory, after the theoretical step between solar system formation and interstellar travel which is causing the problem, and thus "filtering out" what would be a galaxy full of intelligent life forms.
The second group of related explanations argue that while intelligent life should be plentiful, some process is eliminating that life before we can make contact with it. One of the most chilling probabilities is that some sort of super-predator species is prowling the galaxy, hunting out and destroying other life forms. Some science fiction writers have speculated that an intensely xenophobic alien race might even be exterminating every other intelligent race it comes across in sort of cosmic genocide. Less sinister, and probably more likely, explanations are that intelligent species simply are not very good at looking after their own survival, and tend to destroy their capacity for interstellar space travel, either by destroying themselves in a nuclear war or by exhausting their planetary resources and sparking devastating climate change.
The third and final set of explanations for the Fermi paradox relate to the limited nature of our observation technology. In short, perhaps the galaxy is full of alien life, just as our models suggest - but we simply lack the ability to see any of it. If, in fact, it turns out that there really is no technological "trick" by which spaceships could get past the speed of light, then unless two alien races happen to evolve literally in each other's backyard, in interstellar terms, communication and interaction would require hundreds or thousands of years of back-and-forth travel.
Still, even if faster-than-light communication and travel is possible, we might not notice. A civilization advanced enough to send ships between stars might not bother with such apparently primitive technology is radio signals - or they might use radio waves, but in a form we don't recognize as unnatural or noteworthy. Less plausible though not theoretically impossible are some even more exotic possibilities. For example, perhaps we do not realize we are surrounded by life because other civilizations don't want us to know it. We might have been isolated in a sort of cosmic nature preserve, or zoo. Perhaps, as paranormal researchers and conspiracy theorists have been insisting for decades, they are actually among us as we speak, but for one reason or another are choosing not to make themselves publicly known.
All in all, the simplest explanation for the Fermi Paradox still is that we have not found ET either because he/she is not out there, or because we have not been searching long enough. However, we still know very little about our galaxy, and there are many possibilities.