The Fermi Paradox states simply that if there are aliens, where are they? It can be seen most elegantly stated in the Drake Equation, where, simply paraphrased if there has been a certain amount of time, and there are so many stars, and so many planets capable of life, and so much life, and so much intelligent life, and a certain percentage of that gets off the planet, then why haven’t we met any aliens. You can plug in any numbers you like in most of those holes, because we just don’t know the answers. Planets capable of life might be much rarer than we think, and intelligent life might be much rarer. You can’t fiddle with time, but you can get a quite satisfying answer to the Fermi Paradox at any of those other link levels. We only have one example of an intelligent species, and we’ve only barely got off the planet and haven’t even got out of our own solar system yet.
The thing that makes the Fermi Paradox interesting for SF is that like the speed of light, you have to have an answer for it. It can be any answer you like, but it has to answer it. This has led to some lovely creative possibilities and, unlike the Singularity, seems to me to be a constraint on the genre that encourages positive things and fascinating speculation.
So, there’s Ken MacLeod’s Engines of Light solution—the aliens are right here, they’re just keeping a low profile. We don’t see any evidence because they don’t want us to. This is also the position of Roswell conspiracists and so on. They’re here but hiding.
A little further out, there’s “they’re nearby, keeping an eye on us, but leaving us alone for their own benign reasons.” That’s Heinlein’s Have Spacesuit, Will Travel—the aliens are vastly more powerful than we are, and they’re leaving us alone until we’re more ready to join them. They don’t want us until we graduate. We don’t see any evidence because they’re so much more advanced. You see this also in Carl Sagan’s Contact and in Clarke’s 2001. This allows for stories where we later join them and have lots of planets to leave alone, as in the Star Trek Prime Directive, and Lloyd Biggle’s Still Small Voice of Trumpets universe.
Then there’s “Earth is a preserve but aliens aren’t so wise.” In David Brin’s Uplift books the universe is full of aliens who have each uplifted other species to full sentience. Planets are gardened for intelligent species, and Earth is lying fallow ready for the next such, while we evolved by mistake—or did we? There’s also a short story—author forgotten—where there are planets full of mindless human-like meat animals, whose alien owners come back to harvest them now and then. There’s also Jerry Pournelle’s Janissaries where aliens are quietly recruiting groups of humans from Earth for very dubious projects.
Related to this is the fairly common idea that they’re just too alien to be interested in us unless we bother them.
These are pretty good answers because they’re inarguable. I mean if they’re there and they’re hiding, of course we see no evidence. QED. But there’s something a little unsatisfying about “we don’t see them because they’re cheating.”
There’s half a ton of SF where the aliens just happen not to have got here yet and could wander by at any moment. Larry Niven’s Known Space is like that, more power to its Outsiders. Le Guin’s Ekumen is also like that. Also, there’s no FTL travel, and crawling along takes time—it’s surprising how few SF universes there are with no FTL or slow FTL. There are generation ships, but not much like the Ekumen where it takes ten years to get anywhere.
Then there are books that break one of the links of the Drake Equation. C.J. Cherryh plays with the “habitable planets” link in her Alliance/Universe novels—very few suns have planets that can support life, humans live in space stations around inhospitable planets. When they do find planets with life, it usually isn’t intelligent, or advanced. (Later she connected these books with the Chanur books where there are lots of aliens, with no explanation yet.) Bujold’s Miles books break the link at intelligence. Life is common, intelligence is vanishingly rare. After all, there are lots of animals as smart as a dog, there’s only one as smart as people.
Vinge’s Zones answer the Fermi Paradox without ever needing to directly address it—if the interesting part of the universe is a long way from here, of course they’re not here! And in Marooned in Realtime he uses the Singularity as an answer--if aliens all have Singularities and go away, that also fits.
In Saberhagen’s Berserker books, technological civilizations destroy each other. They aren’t here because they’re all off fighting elsewhere, and when they show up, watch out.
Perhaps my favourite answer to the Fermi Paradox is Terry Bisson’s “They’re Made Out Of Meat”.