These are in no particular order, but I think I might list those that I felt most moved by, in one way or another, at the top.
Solanin- Inio Asano delivers a painfully relatable punch to the gut through exploration of the ennui experienced by a twenty-something office worker who leaves her job to try to discover what makes her happy. You should also consider What a Wonderful World 1&2, a series of vignettes that explores some similar themes to Solanin.
Pluto- Naoki Urasawa and Osamu Tezuka- This is a retelling of a classic AstroBoy tale, but rather than take 40 pages or so, Urasawa takes 8 volumes. It's a murder mystery that's an exploration into civil rights, the justifiability of war, and how we define our humanity. Urasawa is a king of the modern manga scene and you really can't get much better than letting him play around with Tezuka's classic. There is a very good chance that you will cry.
I guarantee there is more manga, but my exposure to it is limited. Check these out and if you like them, you've got a pretty diverse rabbit hole to tumble down into.
Okay, let's get the big American comics out of the way. Sandman, Watchmen, Dark Knight Returns and Batman: Year One are pretty much the birth of most of modern comics. Alan Moore's tenure on Swamp Thing probably deserves to be there as well as Morrison's Animal Man. They're all great. You should give them all a shot at some point, but everyone is going to recommend them to you, so I'm going to bypass them.
Let me tell you about King City, because holy shit does Brandon Graham know how to make great comics. King City is a 12-issue series that focuses around Joe, a burglar, and his cat, Earthling J.J. Cattingsworth III. Earthling is a MacGuffin, able to be injected with a vial of cat juice to do pretty much whatever Joe needs, which allows Graham to explore this quasi-sci-fi setting from the perspective of a guy just trying to do his job.
What makes King City so great, though, is that it accomplishes in comics what poetry attempts to accomplish with words. Sure, there are story arcs and character developments, exciting battles and painful heartbreaks, but somehow Graham is able to mesh words with pictures in such a way as to convey a feeling, rather than simply an idea. It's got a $20 MSRP and is over 400 pages of comics. You likely won't find a better deal and I have yet to meet anyone who has disliked it.
Essex County- Jeff Lemire has won a ton of awards for this, and they are all deserved. It's a look inside a small county in Canada and the lives of a couple of families and the county nurse. It's moving, the artwork is beautiful, and like Graham, Lemire knows how to let the art breathe, oftentimes serving to display the vastness of the snow-encrusted forests and the loneliness of childhood in these areas with unprecedented vividness. If you like this and really dig dystopic fiction, check out Sweet Tooth, from Vertigo.
Unlikely- Jeffrey Brown is my go-to for autobiographical comics. There are a lot of people who would recommend Clumsy as it was his first published work, but many of the people I know who have read it struggle to get past his style, which has evolved considerably since then. I feel like Unlikely is still rough enough to seem amateurish in style while delivering a beautifully painful story. It's the story of Brown's first real relationship and really showcases his talent and willingness to be completely open with the reader.
If the idea of reading something about relationships isn't your thing, that's fine, don't give up on Brown. Funny Misshapen Body is the story of how he came to become a cartoonist and was the first thing of his I read. It's great and wonderful, but a little harder to get and you won't see as many reviews floating around of it.
Also, here's Brown's answer as to why most of his comics aren't in chronological order:
American Elf- James Kochalka has done a daily comic strip for the past 13 years. There's more to it than that, but you require no monetary investment, as it's all online at www.americanelf.com. I have yet to see a better representation of the joys and mundanity of life.
Asterios Polyp- David Mazzucchelli takes the medium of comics and bends it over his knee. He does things with shape and color that I have never seen anyone do before and this graphic novel deserves all of the praise it has received and then some. This is everything you should ever want in a comic. The New York Times gave it a glowing review.
The Invisibles- Grant Morrison's counterculture epic that inspired/was stolen by the Wachowskis for elements of The Matrix. Seven trade paperbacks that scream to be read until they fall apart, not just because they are great and exciting, but because they are full of so many ideas that you will mine new ones each time. There have been two prose books examining the ideas behind The Invisibles, and there could probably be more.
Lucifer- Mike Carey writes a 75 issue spin-off of Sandman, exploring a Lucifer who has given up ruling Hell. If you have even the vaguest interest in Christian mythology, this is for you. It's also for you if you liked Sandman but wanted a more direct narrative.
I have carefully avoided traditional superhero stories. It's not that they aren't good, there are some truly spectacular ones, but there are also lists of those. My personal favorites are All Star Superman, Animal Man (Grant Morrison's run), Jack Kirby's Fourth World Saga, pretty much anything by Mark Waid and a bunch of things I'm forgetting at the moment.
Then there's a mess of indie stuff I haven't read or haven't listed that I can tell you has earned its reputation. Cerebus, Blankets, Ghost World, My Friend Dahmer, Bone, I Kill Giants, anything by Chester Brown or Harvey Pekar, Atomic Robo, Big Questions, Finder, Love and Rockets, and the list goes on. I've omitted Transmetropolitan and Y: The Last Man because the requestor had already read those, but yeah, go read those too.
But this is a start.