, on Disney+ Caution: spoilers!
Season One
This is set in the Star Wars universe, five years after Return of the Jedi. The Empire has mostly fallen, though there are still remnants of it scattered around. And there's still plenty of lawlessness; I don't recall hearing any mention of the New Republic, but they clearly haven't gained control in every star system. Anyway, there's a Guild of bounty hunters, led by Greef Karga (Carl Weathers) on the planet Nevarro. He sends a Mandalorian bounty hunter on a job to find an unknown 50-year-old being, who turns out to be an infant of the same species as Yoda. (It's kind of strange that no one on the show has any idea what species he is, but I suppose one can't know every race in the galaxy, and this particular race is probably somewhat reclusive. What's much stranger to me is that most people have apparently never heard of Jedi.) When the bounty hunter (whom people just call "Mando") finds the target, so does another bounty hunter, an IG-11 droid who teams up with him... but their partnership is short-lived. Mando ultimately decides not to turn over the Child, to Greef or his client, which leads to Mando being exiled from the Guild. Throughout the season, he has to protect the Child from bounty hunters as well as Imperial forces. He does make a few allies throughout the season, including Cara Dune, a former shock trooper of the Rebel Alliance, who is now a mercenary. There's also an Ugnaught named Kuiil (voiced by Nick Nolte), whom Mando helped with a problem. He later finds, repairs, and reprograms IG-11 after Mando had shot the droid and left it for dead when it tried to kill the Child. At the end of the season, Mando teams up with Kuiil, IG-11, and Cara, on a mission to kill an Imperial officer who has taken over Greef's town, I guess. If they do this, Greef says Mando will be reinstated in the Guild. Unfortunately, things get far more complicated than expected when a ton of stormtroopers (and better soldiers) show up, led by Moff Gideon (Giancarlo Esposito). He knows a great deal about everyone on the team, and about the Child. What he wants with the Child remains a mystery, but he clearly wans him very badly. In the end, Mando, Cara, and Greef manage to escape, but I'm sure we'll see more of Gideon next season. For now, Mando is tasked (or "quested") with learning where the Child comes from so he can deliver him to his people. But he'll also want to find his own people, as the Imperials had killed a bunch of Mandalorians, but it's hoped that some may have gotten away, offworld.
Well, I'm leaving out lots of details, as there are plenty of exciting adventures on various worlds and in space, throughout the 8-episode season. I'm just not sure right now how important any of it is to the overall story arc of the series, so I don't know what else to tell you. (Except, of course, that the Child became quite popular online, with countless memes and things, in which he's often referred to as "Baby Yoda," though of course that's not who he is.) Anyway, I'm sure I'll have more to say next season.
Season Two
First, I should say that I originally thought Mando would have to try to find more of the Child's species, but it turns out he's actually searching for Jedi. In the third episode, Mando meets a group of Mandalorians who don't follow the same religion as he does, which means unlike him, they can take off their helmets in front of people. The leader of the group is Bo-Katan Kryze (Katee Sackhoff), who wants to retake Mandalore from the Empire. (She's been seen in other Star Wars media before, but I don't think I've seen/read about her before this.) She also tells Mando where he can find a Jedi, and in episode 5, he meets Ahsoka Tano (Rosario Dawson). Her I did already know, from the animated series The Clone Wars, though this was the first time she appeared in live-action media. (She had also later appeared in Rebels season 2, which I hadn't seen yet when I watched this.) She communicates with the Child, and tells Mando his real name, which is Grogu. (Speaking of which, we had also learned Mando's real name at the end of season one, I didn't want to spoil it at the time. But now I'll say that his name is Din Djarin, though I'll continue calling him "Mando".) Um... anyway, Ahsoka tells him about an ancient Jedi temple on planet Tython, where she says he should take Grogu, so he can reach out with the Force to try to contact another Jedi. (Ahsoka couldn't take Grogu herself, for reasons.) While on Tython in episode 6, Mando meets Boba Fett (Temuera Morrison), who had been assumed dead ever since "Return of the Jedi". He's now working with someone else who had been assumed dead, an assassin named Fennec Shand (Ming-na Wen) whom Mando had fought in one episode of season one. Moff Gideon blows up Mando's ship (the Razor Crest, which I guess I hadn't mentioned the name of before), after which Mando begins traveling with Boba and Fennec on Boba's ship, the Slave I. They agree to help him rescue Grogu after Gideon abducts him. (Actually, he was abducted by some Dark Troopers, which we later learn are some very badass droids). They're aided in the rescue attempt (in episode 8) by Bo-Katan and one of her Mandalorian allies, Koska Reeves, as well as Cara Dune (who is now a marshal of the New Republic). Cara wants to capture Gideon for the Republic, while Bo-Katan wants to defeat Gideon in order to claim the Darksaber he wields, which would prove her the rightful heir to the throne of Mandalore. Well, Cara gets her wish, but Bo-Katan's desire doesn't quite go as planned. (I won't say why, though I'm sure it will come up again in season 3.) Anyway, they all probably would have been killed by the Dark Troopers, if not for the fact that a Jedi finally shows up, having answered Grogu's call. He pretty easily defeats the droids, and takes Grogu into his care.
There's a post-credits scene the nature of which I don't want to spoil, but it sets up a spin-off series called The Book of Boba Fett.
Season Three
Well, I suppose the first thing I should say is that after spending some time training with none other than Luke Skywalker, Grogu returns to the Mandalorian by the end of "The Book of Boba Fett". Also in that show, Mando got a new ship to replace the Razor Crest, a refurbished N-1 Starfighter, which I think he just refers to as "N-1", IIRC. And this season on "The Mandalorian", Mando gets an astromech droid called R5-D4. And plenty of stuff happens that I'm not going to go into, but the important thing is that at one point, Bo-Katan earns the right to wield the Darksaber, and she works with Mando to reunite different tribes of Mandalorians, and they attempt to retake Mandalore, which Moff Gideon (who had escaped from New Republic forces) is now using as a base of operations. In the process, the Darksaber is destroyed, but I think Bo-Katan will remain the leader of the Mandalorians, anyway. Later, Mando approaches a New Republic captain named Carson Teva, whom we had seen a few times before, and asks him to hire Mando on a case by case basis, unofficially, to deal with various problems. And Mando adopts Grogu, who is now known as Din Grogu, and they get a home on Nevarro.