The Fermi Paradox is the apparent contradiction between the lack of evidence for extra-solar civilizations and the fact that there is a high probability that many Earth-like planets exist that are older than ours and that it would conceivably be possible to explore the galaxy in a couple million years. Simply stated, where are the aliens because they should be here. There are many hypotheses for why this may be the case. Some deal with the nature of civilizations, the nature of the universe, and even conspiracies the aliens have been found but the government is hiding this from us. Here is a list of all the hypotheses that Wikipedia summarizes:
One of the ones I find most interesting is the hypothesis that they are too alien. This one states that the difference between us and other life could be so extreme that communication is all but impossible. Perhaps these aliens think and communicate so much slower than us that we would mistake communication for background noise. Or perhaps these aliens understand physics and math completely differently.
In general, I find the fact that the life on Earth is the only one that we can base any hypothesis on to be most limiting factor in any search for extra-solar civilizations. Given that we have no other examples of life to base our thoughts on, our search for life is greatly inhibited by this lack of understanding. Additionally, we don’t even understand how we got here in the first place, what lead to eukaryotic cells and sexual reproduction, thus it is pretty difficult to postulate what other civilizations would be like. However, these limitations certainly don’t mean we should stop searching. In fact they are motivations to continue searching because perhaps an answer to life elsewhere would lead to answers about the nature of life of earth.