Also see: Drake Equation, Intelligent aliens, Speculative civilizations
Fermi Paradox was first pointed in 1950 by the physicists Enrico Fermi and Michael Hart based on probabilistic arguments about the existence of alien civilizations, such as the Drake Equation: if there really is a great amount of technologically advanced extraterrestrial civilizations, as estimates seem to predict, why until now haven't we seen any direct evidence of the existence of even one of them? This becomes even more problematic if we consider that several star systems, and even an entire galaxy, could be colonized in a few million years, a very short time that should have already passed: if even one advanced civilization ever existed before us in the Milky Way, we should expect the entire galaxy to belong to it by now.
Hypothetical solutions
There are no other civilizations
It's entirely possible, of course, that probabilistical arguments in favour of intelligence in the cosmos are simply wrong. There is a great uncertainty of the actual probabilities of each passage: planets suitable for life might be much less common than we hope (see the Rare Earth Hypothesis); even if those planets are common, the origin of life might be unlikely; even if it's not, life might be very susceptible to mass exinctions, and disappear easily or fail to develop intelligence; even if intelligence is common in the Universe, intelligent species might fail to develop highly technological civilizations, dying out earlier or existing in a relatively primitive way (after all, even today several cultures on Earth have mantained their traditional technology since Paleolithic).
According to a somewhat more somber argument, technological civilizations might in fact be common, but they could have a tendency to destroy themselves either indirectly, through pollution and resource depletion, or directly, through war or accidental technological disasters (e.g. the grey goo scenario). This view was held by several scientists, such as Carl Sagan, during the Cold War; some suspect that any intelligent civilization is bound to be on some measure aggressive and xenophobic, as these traits would be naturally produced by evolution. Besides, intelligent species tend to occupy a higher trophic level in their ecosystem.