Space is big. Really big. You just won’t believe how vastly hugely mind-bogglingly big it is.1
Pondering this fact, in 1950 Enrico Fermi over lunch posed the now famous question concerning extraterrestrial life: "Where is everybody?" He was right: Considering the size and age of the universe we should be up to our eyeballs in aliens. Where the hell are they?
Leaving out for a moment the size of the universe, let's just focus on our own galaxy, the Milky Way. Its diameter is (generously) somewhere between 150,000 and 200,000 light years and it contains up to an estimated 400 billion stars.2
Those are ridiculously huge numbers (one might even call them astronomical), but even without any faster-than-light means of transportation, an intelligent, space-faring species should be able to explore a galaxy like our own within as little as a million years (using self-replicating probes)3 and full colonization could probably be achieved on a similar time scale,4 but certainly in far less than 100 million years.5
In terms of the age of the galaxy, that's really nothing. The Milky Way has existed in roughly its current form for about 10 billion years. That would have been enough time to settle and/or explore it in its entirety thousands of times.
Since life evolved on earth (and earth does not seem to be special), the assumption is that it should reasonably also have evolved in any number of other places, some being billions of years older than earth. This assessment gains even more weight with the recent discovery of a large number of planets outside our solar system.6 The realistic estimates for earth-like planets in the Milky Way alone currently stand at more than 10 billion.7 If anyone intelligent was around in all those billions of years on all those billions of worlds and if they have the same inclination to expand their habitat as all life on earth, it's reasonable to assume that by now they would be everywhere - including here on good ol' earth.
This in essence is the Fermi Paradox: From the evidence we can see, there ought to be a large number of extraterrestrial life forms all over the place. But there aren't. What gives?
There Is No Life
Obviously, there has to be a rational explanation for the observed discrepancy between likelihood of intelligent life and our failure to encounter it in our environment. Over the years, quite a number of solutions to the Fermi Paradox have indeed been proposed.
One apparent explanation of course is that life really is just so rare that humanity is the only species that so far has "won the lottery of life" and survived all the way (or at least almost) to being a space-faring civilization.
The idea here is that at least one evolutionary, societal, or technological step from dead matter to settling the galaxy is so hard that it can almost never be achieved. We call such a step a Great Filter and it could for example be the step from single to multi-cellular life, from animal to intelligent life, or the inevitability of destruction by anything from warfare to asteroid impact to gamma ray burst.
Looking out into the future, it's almost certain we will find life that's not like us out between the stars: once humanity starts settling the Milky Way, local societies should become enough isolated and adapted to their local environments that they will become unable to interbreed. At that point, mankind will probably split into a large number of related, but different life forms.
There Is Life
On the other hand, maybe there is a lot of life, but we are just not looking in the right spot or with the right means. A civilization millions of years ahead of us might communicate using a technology we don't have, their communication might be encrypted (looking like noise), or they may be around, intentionally isolating us, or we simply might not be able to detect them (or their probes).
From our past observations of the heavens we are pretty sure that everything we see in the universe is of natural origin - there don't seem to be any major artificial structures8 or unexplained influences on natural phenomena, although we had out doubts in the past.910
Then again, maybe everyone else out there in the universe is wise enough to stay safe from others by shutting up and not attracting attention. Or maybe, the universe is not real at all, but a simulated reality, used for example for entertainment, training, or research. If that's the case, there really might be a god, or - more likely - a snot-nosed IT nerd watching your every hilarious misadventure. Smile and look into the camera.