Along with the likes of Evangelion, this is one of the anime that defines the medium. Like it or hate it, you can't ignore it. This is not a show for everyone, and judging by a few of the negative reviews here, some people flat out don't get it. So let's set a few things straight.
One: It is not a masterpiece. It's a flawed, weird, clumsy, uneven mess of a show.
Two: It is wildly original. There are many legitimate criticisms of this show. A lack of originality is not a even close to being a fair one.
Three: It is very well made. But we expect that from KyoAni. Even when they make trash like K-On, it's exceptionally well made trash.
Sometimes works of fiction are named after a single character. In many cases this is the main character and in many cases that character is not really special enough to merit having the whole damn thing named after them. The Harry Potter series is the most successful series of novels of all time without competition, yet they are named after a fairly passable, but unexceptional main character. This show is not like that. Haruhi is a character that demands attention. Placing her name in the title is beyond well-earned.
So let's talk about her.
She is objectively a bad person. She also poses a cosmic threat to the universe itself with her near limitless powers. Despite this though, she is a wildly compelling character. Her horribleness never spills over into outright villainy and she occasionally even does noble things. Trying to decide what to think of this enigma of a human being/cosmic deity is one of the core elements driving engagement with this show. I still don't really know how to judge her. But I can't help but be swept up in her bizarre charm. She is a character I've truly never seen before. A wildly simplistic analysis might compare her to the Manic Pixie Dream Girl phenomenon, but anyone who considers Haruhi a dream girl has some real issues to explore. She is narcissistic, often cruel, a sexual predator, and utterly amoral. Some might consider her a Mary Sue - she's (at least in some ways) smart, attractive and athletically gifted, and yet her deficiencies as a human being are gargantuan.
What is she? She's Haruhi Suzumiya, nothing more, nothing less, nothing else.
The other characters are more of a mixed bunch. Kyon, our narrator, is a brilliant foil to Haruhi and it's hard to imagine the show working without his droll, understated outlook backing it up. Kyon is a character that is absolutely perfect for his role in the show in a way few characters are.
The rest of the SOS Brigade isn't quite as distinguished but Nagato and Asahina fulfill their roles with aplomb, nonetheless. One feels generally sorry for the latter on the regular, thanks to Haruhi's unending abuse. These characters depend heavily on the more fantastical elements of the show to flesh them out.
Overall, this is a wonderfully animated show, full of a great characters, great music and a confusing, yet decent plot. But let's not kid ourselves: you watch this for Haruhi and that is in itself enough. Just enjoy our capricious goddess.