It's the Sword Art Online anime no one knows about. (I didn't, anyway, despite having followed Sword Art Online for a decade.) It's by the same author (Reki Kawahara) and set in the same fictional universe, but a generation later and there are no crossover characters. The tone's different too.
This series came out the same year as Sword Art Online's did, but that exploded into what's now one of anime's major franchises. Accel World is, though, arguably better. Sword Art Online's main advantages are: (a) it was literally life-and-death, especially in its first season. It was scarier and had the dramatic weight of real death. (b) it's cooler, which I suspect was the real reason for lots of viewers. Kirito is cool and always wins, while Accel World's Haru looks like a loser and sees himself as one.
It's about teenagers playing a game. It's the year 2065 (although only 2045 in the original light novels), i.e. 43 years after the start of Sword Art Online. Haruyuki Arata (aka. Haru) is a short, fat boy who gets bullied at school and whose online avatar is a cartoon pig. Good grief, he's tiny. From some angles, he looks spherical. One day, though, an aloof beauty at school (Kuroyukihime) takes what normal people would call an incomprehensible interest in him. Kuroyukihime is an unhealthily dedicated gamer, you see, and she finds Haru attractive for his gaming prowess. She invites him to participate in a game called Accel World, which is short for "accelerated world" because playing it speeds up time 1000-fold.
What's interesting is how the Accel and real worlds are tied together. Firstly, your game avatar and its powers will reflect your deepest desires. The game's embodying its heroes' characterisation. Secondly, game losses can have permanent effects. If you get reduced to zero points, the game uninstalls itself irreversibly from your brain while deleting all memories associated with it. (That last bit means I'd never install it. Imagine getting addicted to a game, committing years of your youth to it and then having all that erased from your memory.) Thirdly, having access to 16 minutes' worth of mental activity in a second can have real-world advantages.
Also, the game's rules mean that even strong characters aren't safe. When spending your points, do you choose promotion or safety? Someone who levels up too fast might get knocked down to zero in one unlucky fight.
Theoretically, the storyline's pretty much what you'd expect. The Accel World has its own aristocracy (super-powerful, don't mess with them), secrets, culture and villains. Haru grows through hard work, training and of course duels. The most amazingly smug antagonist shows up halfway through and you'll hate him like poison.
In practice, though, it kept being better than I expected. The fights all have a dramatic point and are saying things about the characters (as well as being exciting) and you couldn't really delete any of them. Haru is a surprisingly memorable hero, since he looks like a loser and his self-esteem issues are initially crippling (which is an obstacle in his relationship with Kuroyukihime, given his perception of their relative statuses). He's capable of making her cry with his assumptions about himself and her. "There's no one in the world I believe in less than myself." This makes him relatable. At the same time, though, he's clever and even valiant.
I prefer the show's first cour (adapting novels 1-2) to its second (novels 3-4). They're both good, though, and I'd be keen to see the show resurrected. Its heroes are much more flawed and interesting than those of Sword Art Online, for a start, while the light novels are still running and have over twenty volumes of material yet to be adapted. It's exciting and often funny (e.g. the contrast between the Red King's "cute little girl" public persona and her thuggishness underneath), while I think Reki Kawahara is an underrated writer. (Although it's weird to say that about someone so successful.) There's nothing particularly amazing about his storylines, but he keeps hitting every beat well. He finds the emotional core, and the weight, and his characters' purpose. Whatever he does next, I expect to watch it.