In 1950 physicist Enrico Fermi was having lunch with his colleagues at Los Alamos when he suddenly asked something like "Where are they?" or "Where is everybody?" (recollections differed). Everyone laughed, because while the question came out of the blue, they had earlier been talking about UFOs and a recent flying saucer cartoon. Fermi was asking why we don't see aliens.
I've heard people say that the vast distances between stars rules out interstellar travel, and in fact what we know suggests that there are very serious obstacles even to traveling at a substantial fraction of the speed of light, let alone faster. That's probably true, but as Fermi et al worked out, even relatively slow-moving spaceships or self-reproducing robot probes could plausibly colonize the galaxy in under a billion years and perhaps a lot less. (For more, search the web for "Fermi paradox" and you'll find a huge amount of information on the subject.)
In fact, we humans have already launched at five interstellar spacecraft (Pioneer 10 and 11, Voyager 1 and 2, and New Horizons. (See this earlier post for more about them.) While they will long since have ceased to function by the time they reach another star system, many tens or hundreds of thousands of years from now, they'll get there eventually.
So where is everybody? One possibility is that we might be among the first technological civilizations in our galaxy. The Earth is about a third the age of the universe (quite close to exactly that, in fact). The Big Bang produced hydrogen, some helium, and a bit of lithium, but the heavier elements were created later in stars, and those heavier than iron produced only by supernovae. The earlier a star or other body formed, the less it had in the way of heavier elements.
Moreover, while life seems to have appeared about as soon as the Earth could support it, it took a few billion years for multicellular life to develop, and technology more advanced than termite mounds is very recent indeed. If we can infer anything from our one solitary example, maybe it just takes a long, long time for intelligent technological life to develop, and we could be the first, or at least among the first.
Here's a short, fascinating discussion of the idea from PBS Studios:
(Updated 2015 Nov 14 to add a few details.)by