Supergirl, on CBS (s1) / The CW (s2-6)
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Caution: spoilers.

I'm not really familiar with Supergirl from the comics, but I've seen different portrayals of her before. There were the 1984 movie Supergirl, the animated series Superman (and other shows in the DCAU), the live-action series Smallville, and possibly other things. But I've been looking forward to this new series, for the same reasons I look forward to any new comics-based movie or show. Also I was interested because Supergirl is being played here by Melissa Benoist, whom I liked in seasons four and five of Glee. Also, some of the show's writer/producers are also involved in making Arrow (and related shows), on the CW. So there was some speculation as to whether this show would become part of the "Arrowverse." (Most of the shows in that universe are on the CW, which is partly owned by CBS, so a crossover didn't seem impossible.)

Season One
In the pilot, we learn that when she was 13, Kara Zor-El was given a mission by her parents, Alura and Zor-El, to go to Earth and protect her cousin, Kal-El, who was a baby. (The two of them were presumably the only survivors when their home planet, Krypton, was destroyed.) This was surprising to me, because I don't recall ever hearing of Kara being older than Kal-El. However, that unfamiliar plot point is soon eliminated by the fact that Kara's escape pod was knocked off course, and she ended up spending 24 years in the Phantom Zone, where no time passes. (It also seemed weird to me that the Phantom Zone was like... just some weird area of space, as opposed to another dimension, but whatever.) Her pod eventually gets out of the Phantom Zone, though, and Kara finally arrives on Earth, still 13 years old (because of the "no time" thing), but her baby cousin has been there for 24 years, so he's an adult now, and already a famous superhero (Superman, whom we have not yet actually seen on the show). He brings Kara to a couple, Jeremiah and Eliza Danvers, scientists who had helped Superman learn to use his own powers, or whatever. (They're played by Dean Cain and Helen Slater, which is a neat casting stunt, because they played Superman on Lois & Clark and Supergirl in the 1984 movie, respectively.) So they'll raise her, along with their own daughter, Alex, and Kara takes on their surname. Since Kara's original mission of protecting Kal-El is no longer necessary, she decides not to use her powers, and just try to live a normal life.

After all this backstory is out of the way, the show flashes forward to the present, when Kara is 25. She now lives in National City, working as an assistant to a media mogul named Cat Grant (Calista Flockhart), the founder of CatCo Worldwide Media. Kara is friends with a coworker named Winslow "Winn" Schott, Jr., who apparently has unrequited romantic feelings for her. And in the pilot, James Olsen (who used to work in Metropolis, at the Daily Planet) moves to National City and starts working at CatCo. He's a friend of Superman (who James says is one of the few people he still lets call him "Jimmy"), and immediately befriends Kara. Then one night, a plane that Alex is on nearly crashes, so Kara decides to use her powers for the first time in years, to save her sister (and all the passengers). This makes the mysterious super woman a media sensation, and leads to the possibility that Cat might not have to downsize her newspaper, the Tribune, as she had planned. It's also Cat who dubs the new superheroine "Supergirl," which Kara objects to, believing it should be "Superwoman." But of course she has no say in the matter, because she can't tell Cat that she is the woman in question. Though she does tell Winn, and it turns out that James already knows. (It's also important to note that the show tries to be feminist, with mixed skill. Cat's explanation of why "girl" isn't problematic is somewhat dubious. And... in general, I'd just like to echo what the A.V. Club says about the pilot.)

Anyway... Kara is really excited to finally be using her powers. (I'd say that even before that, she's kind of adorkable, but her enthusiasm makes her even more so.) However, Alex is upset about it, and wants her to stop, which briefly leads to a rift between the sisters. But we soon learn that that's because Alex is part of an organization called the DEO (Department of Extra-Normal Operations), which monitors and combats alien threats to Earth. And there's a group of alien super-criminals who had been in a prison ship called Fort Rozz, in the Phantom Zone, which came to Earth at the same time as Kara, and its prisoners all escaped. since then, some of them have been captured by the DEO, while others remain at large. The head of the DEO is Hank Henshaw, who doesn't trust any alien (including Superman and Supergirl), though Alex convinces him to let Supergirl work with them. We eventually learn that Alex's father, Jeremiah, had secretly worked for the DEO, and was killed on a mission, years ago. And that Henshaw was the last person to see him alive, and had kept the secret from Alex. So she and Kara begin to distrust him. However, Alex eventually learns that the man she's been working for isn't the real Hank Henshaw. (I guessed his true identity almost from the beginning of the series, but it wasn't revealed until episode 7.) He's actually a Martian named J'onn J'onzz, who is the last of his race. He also has various super powers, one of which is shapeshifting, which is how he assumed Henshaw's identity. I don't want to spoil all the details concerning that, but suffice to say he had made a promise to Jeremiah to look after his daughters. Anyway, Alex now realizes she can trust him... but she can't reveal the truth to anyone, including Kara. (Though it's not too long before Kara learns the truth, too.)

Meanwhile, one of the Fort Rozz escapees is Kara's aunt, Astra, the identical twin of Alura. Astra is the general of a group of former alien prisoners (some Kryptonian and some of other races), including her husband, Non. Astra claims she had tried to save Krypton from destruction, but Alura and the other rulers refused to listen to her (largely because she resorted to terrorist methods to try to save the world), and now she wants to prevent a similar catastrophe from occurring on Earth (employing similar methods). Before Kara left Krypton, she was close to her aunt, but now they become enemies. And there are various other characters I should mention, including a genius technology mogul named Maxwell Lord, who becomes a potential threat to Supergirl (though so far the public is unaware of any wrongdoing). He kind of reminds me of Lex Luthor. And there's a woman named Lucy Lane, who was formerly involved with James, but broke up with him before he moved to National City. She eventually comes to the city, and rekindles her relationship with James, which makes Kara jealous. Lucy is also a lawyer and a major in the U.S. Army, under the command of her father, General Sam Lane, who distrusts aliens like Superman and Supergirl.

There's also an episode where we learn that Winn's father is a criminal called Toyman, a fact of which Winn is ashamed. Also, Kara briefly dates Cat's son, Adam. And eventually, Astra is killed, though I won't spoil how that happens. Non takes over leadership of the Fort Rozz criminals after that, and continues to work on implementing Astra's plan, which involves something called "Myriad." And Cat gets another assistant, Siobhan Smythe, who develops a rivalry with Kara, but also starts a relationship with Winn. (Though neither her position at CatCo nor her relationship last very long.) And... in the comics, there is a villain, or rather a series of villains, known as Brainiac. (Actually, at least one incarnation of Brainiac was a hero.) And in this series, Brainiac 8 (aka Indigo) becomes an ally to Non. (She's played by Laura Vandervoort, who had previously played Kara on Smallville.) And in episode 18, there's finally a crossover with the Arrowverse, when The Flash winds up in National City and helps Supergirl against a couple of villains. However, this show doesn't exactly become part of the Arrowverse, because the Flash only arrived by experimenting with interdimensional travel. (I had been missing the second half of the current season of all the Arrowverse shows, so I was worried this crossover might spoil some things, but I don't really feel like it did.) And... in the final two episodes of the season, Supergirl has to try to stop Non and Indigo's plans. She does so with some help from Cat and, surprisingly, Max Lord. The bad guys are finally defeated, but the finale ends on a cliffhanger, the nature of which I don't want to spoil.

Season Two
The show moved from CBS to the CW, which I don't get, so I'm not sure when I'll be able to see this season.