Initially, Rick and Morty was a parody of Back to the Future's Doc and Marty. It was a poorly animated short, whose solitary joke was that "Doc" explains to "Marty" that the only way to prevent some horrible future catastrophe was for Marty to lick Doc's testicles. It's patently obvious this is nothing more than a sick joke on Doc's part to get Marty to lick his balls. Barely even worth animating. But the voice characterizations were the same: with Doc having the gravelly, angry voice that's used to voice the later "Rick Sanchez", and a parody of Michael J. Fox's tendency to have his voice crack into a higher register was the inspiration for the voice that would later become Morty's.
In order not to get sued into oblivion, "Doc" became "Rick", "Marty" became "Morty", and when the show became its own thing in its own right, instead of a time travelling DeLorean the travel was achieved by a variety of methods. But most specifically, a portal gun, a piece of technology that literally punches holes in reality, allowing for doors to open in space and time, and across parallel universes. Because of this ingenious storytelling "McGuffin", the show in no way has to respect any potential paradoxes of time travel, and the pair have more than once simply aborted an unresolvable fate by simply inhabiting another parallel reality.
In this universe, mad scientist "Rick Sanchez" basically returns to his estanged daughter's life after many years of absence. Because of her daddy issues she's married a sad sack of a milquetoast man, "Jerry" (voiced by Chris Parnell) and fathered two children, Summer (the voice of Frasier actor Kelsey Grammer's daughter) and Morty. Rick installs himself in the garage, turning it into a laboratory, and into their lives - with disastrous results. Their normal, suburban life is no more. Especially not with Rick Sanchez dragging his grandson across galaxies, times, and even parallel universes to have adventures.
The show started out full of early installment weirdness and gross-out jokes. But as the seasons went on, references from earlier shows became very meaningful in later ones. There were shout-outs to other movies and properties everywhere. Although ostensibly based on Back to the Future originally, it owes more to Doctor Who's formula of "cranky old mad scientist, and young naive companion". (As well, by the way, the theme music's counterpoint of menacing chuddering bass line, and ethereal analog synth pad melody) But whereas the companions were and still are there to provide exposition ("What is it Doctor?" "Well, you see, Mel, this is a Zygromic Bergomulator, and what it does is provide us with a convenient deus ex machina at the 27th minute...")...
Morty started out originally as the polar opposite to Rick ("I (burp) need you to cancel out my brain waves so I can't be tracked!). Rick is an alcoholic, amoral nihilist who lives simply for kicks with little regard to the mayhem his adventures leave behind or the human wreckage of broken feelings and hurt people he leaves in his wake. Morty is a sensitive boy with a good moral compass who, despite destroying the Earth trying to get the cute girl at school to make love to him (they turn into "Cronenberg world" so named because everyone resembles the victim of some David Cronenberg mishap) is a naive, simple good soul at heart.
But as the seasons wore on, the interplay between the characters became more meaningful. Rick's return meant that his daughter finally had the courage and sense to dump her useless husband, even though he's dysfunctional and she's still clinging to the hope he'll be a real dad to her. Summer and Morty don't deal with their parents' breakup well, Summer turning into a literal Mad Max wasteland savage with a Mike Tyson facial tattoo.(For the record that episode made a very, VERY nice use of the "returned dead severed hand taking revenge" horror trope). This latest season (Season 3) has the darkest episodes yet, with Rick finally realizing that he hates himself to the point of being unconsciously suicidal and that he truly loves Morty very very much, despite his rough outer shell and protestations that nobody and nothing matters.
What the season has turned into is a very deep examination of the human condition, complete with motifs of life after death (wronged people come back, implying death is not the end), the necessity of the Angel and the Devil in our consciousness, and so forth. In the most interesting episode to date, Rick and Morty rid themselves of their psychological toxins (anger, hate, fear, self-hatred) in alien spa, with intriguing results (they're anthropomorphized as green gooey versions of themselves). Rick is no longer driven, and the "pure toxic" Rick becomes smarter and actually discovers a new kind of matter - implying that madness and genius always co-exist. Morty loses his self-hate and fear and turns into a very disturbing high-earning sociopath with a trophy girlfriend. But intriguingly, bound up in "toxic Rick" is a deep love for Morty, which implies that the passion he has for his grandson transcends normal grand-paternal love. Rick realizes that it's actually dysfunctional to rid yourself of all negative emotion and must instead befriend and integrate and understand your own dark side - an idea many a religion has tried to impress upon its followers.
There are still some unexplained things about the show. We know of a one-eyed "Evil Morty" and a mentally challenged Rick in different universes. Flashbacks during a brain-scan in one show imply that Rick's real "Morty" died along with his entire family, and that this Rick hopped into another dimension to cling to another Morty out of guilt and pain. This would explain why his love for Rick would be seen by the alien spa as "negative". One of the reasons why Season 3 was repeatedly delayed, much to the chagrin of its fan base, is that they wanted to create the densest, shout-out and plot-advancing riddle box yet, and were clearly poring over the show's Bible looking to start tying up loose ends and introducing intriguing new ones.
Despite all this, it's a fucking delightful send up of genre fiction, complete with meta-commentaries on the genre and others. Adult Swim got in on the fun by putting a bumper featuring an in memoriam panel for Game of Thrones writing, 2011-2016.
This of course, has led to fan theories a-plenty, and many a YouTube channel devoted to seeking out and pointing out the many, many obscure references and back-shout outs in each new episode. As the story wears on the characters become more complicated and the questions we have about them multiply and deepen. This isn't simply a fart joke animated show about a crazy old man and a shy kid, it's becoming its own canvas upon which to paint a lot of questions about the nature of the universe we live in, and the potential twists and turns it could take.