A few years ago, at Christmas, someone in my family (I think my mom) got me a pair of Justice League pajama pants, adorned with pictures of Superman, Batman, Green Lantern, etc. (and they're awesome, but that is neither here nor there). This led to my sister and I having the following exchange:
Bethany: "What's Green Lantern's power anyway?"
Me: "He's got a power ring that can make anything he can think of."
Bethany: "A power ring? Why is he called 'Green Lantern' then?"
Me: "Well, the ring has a limited power supply, so he has this lantern to recharge it. And to recharge it, he holds the ring in front of the lantern and says 'In brightest day, in darkest night, no evil shall escape my sight. Let those who worship evil's might beware my power, Green Lantern's light.'"
Bethany: "..."
Yes, folks, I am a tremendous nerd. I am a nerd about many, many things, but one of those things is comic books. I love comics. I spent large quantities of my youth pretending to be a superhero of some sort or another. And its not just superhero comics I love, its the medium itself. Comics, when done well, can tell stories in a way that no other medium can.
Now, as a nerd, I'm all about introducing people to the nerdy things I love. Comics have much less of a stigma now than they did even 5-10 years ago, helped mightily by the popularity of superhero movies. Still, I'd imagine that most people reading this don't read comics. I don't blame you, it's a tough world to dive into. Much of the comics universe (much like any medium) is crappy, and especially the mainstream stuff (again, much like any medium). Not only that, but all the superhero comics have huge, elaborate continuities that are daunting to dive into. So, if you're interested in taking a chance on comics (and you should), I offer my services as a guide into that world. Mostly, I just want to talk about things I love :).
Just as a preface to my recommendations, let me say that it is pretty easy to find comics in your local library. Most libraries I've seen carry a lot of trade paperbacks (collections of individual comics into story arcs), so they're easy to get, and free, so that makes it easy for you. Anyway, on to the recommendations
Non-Superheroes
That's right, there are comics that aren't about superheroes. It's not all about capes and powers. The medium has been used in a lot of different ways, and these comics, due to the lack of huge continuities, are often better introductions to the comics world.
Bone, by Jeff Smith - If you've never read comics, I can't think of any better place to start than Bone. The best description I can give is Calvin and Hobbes mixed with Lord of the Rings. It feels like a simple cartoon in the beginning, like something you'd find in a newspaper strip. However, you soon find that there's a deeper mythology under the surface, and it unravels into an epic adventure full of humor and heart. It's easy to pick up and hard to put down, and accessible to anyone. I can't recommend it highly enough.
Sandman, by Neil Gaiman - Sandman is my all-time favorite comic series, and among the best things I've ever read in any medium. It's entirely unique, with a really spectacular premise and amazing execution. I don't know exactly how to describe it without going into great detail, but I can just tell you that it's absolutely amazing and you should give it a try. The art is gorgeous, and the 75 issue run is full of loosely connected vignettes about the power of stories and dreams. As "adult comics" go, it doesn't get any better.
Maus, by Art Spiegelman - If you want to see the uniqueness of the comics medium, Maus is a shining example. Spiegelman tells his father's story of being a Holocaust survivor, as well as the story of him hearing that story from his father. His brilliant twist is that all the people are portrayed as animals. By anthropomorphizing the people, Spiegelman is able to universalize the story and let the reader insert himself into the story and empathize. It's a brilliant, powerful comic and, oh yeah, it won the Pulitzer.
Y:The Last Man, by Brian K Vaughn - If its an epic adventure you want, Y: The Last Man is a great choice. It's a fantastic sci-fi premise: One day, inexplicably, all the men on earth die, except for one nerdy escape artist and his monkey. As with all great science fiction, Y isn't concerned with the "how?", it's concerned with the "what now?" It's a sweeping adventure, as the last male in the world makes his way across the country and tries not to get in trouble. It's really fantastic, and it sticks the landing with one of the best last issues I've ever read in a comic run.
So, there's a start for getting into non-superhero comics. These run the gamut from light (Bone) to heavy (Maus) and showcase a variety of writing and artwork styles. They're all great, and all worth your time, even if you have no interest in capes and masks.
Superheroes
Now, moving on to the world of superheroes is a little trickier. Where do you start with a character like Superman, who has nearly a century worth of backstory? Well, it's not easy, but I'll do my best to recommend some of the stronger comic runs that are largely self-contained.
Batman: Year One, by Frank Miller - This is the story that really created the modern Batman. The dark, gritty character you see in Batman Begins and The Dark Knight got his start here, in Frank Miller's comic. Miller was one of the best comics writers on the planet for about 15 years, and this is him in his prime. It's also an origin story, so it doesn't get much more self-contained. If the character of Batman interests you, this is the place to start. (also, while we're on Batman, I'll recommend The Long Halloween, another largely self-contained story that is extremely well done and features some great artwork).
Ultimate Spider-Man, by Brian Michael Bendis - Back in 2000, Marvel comics decided that their characters had way too much backstory and convoluted continuities to draw in new readers. So, they relaunched a bunch of their characters in the "Ultimate" line. They started over with them and told brand new stories with new versions of the characters, unburdened by decades of backstory. Most of these titles were not very good. However, in Ultimate Spider-Man, they found a winner. It's a modern version of Spider-Man, told from the origin on, and for the most part it's consistently great. The writer/artist combination lasted the first 110 issues of the run, which is absolutely unprecedented longevity in the comics world. That led to a comic that is consistently strong and has a great handle on its characters. It's a great gateway into superhero comics.
All-Star Superman, by Grant Morrison - All-Star Superman is one of the most fun comics you'll ever read. It's a completely self-contained take on Superman in a futuristic setting, and Morrison uses it as a canvas to tell a bunch of amazing, over-the-top stories that are just pure fun, and a complete encapsulation of what makes Superman a beloved character. Morrison's writing is often overly convoluted, but here it's simple, straightforward, and just a joyous celebration of the character. It's the best Superman story ever written, and it's completely accessible, even if you've never picked up a comic in your life.
Where Not To Start
X-Men: I love the X-Men. Growing up they were my favorite superheroes. However, they do not lend themselves well to new readers. X-Men comics have easily the most convoluted continuity of any comic franchise (which is saying something). There are a million characters, alternate timelines, aliens, and characters who come back from the dead (multiple times!). Frankly, it's a mess. There are some truly great X-Men stories out there (The Dark Phoenix Saga, Astonishing X-Men, to name a couple), but there's also a whole lot of crap, and the long, winding history can make it very frustrating to newbies.
Alan Moore - If you ever look up lists of the greatest comics of all time, you're going to see the name "Alan Moore" a lot. Specifically, you're going to see his comic Watchmen, which is almost universally praised as the greatest comic of all time. Now, don't get me wrong, Moore is terrific. He's a kind of mad, eccentric genius (and I mean that, the guy is an odd fellow) and writes comics unlike anyone else. For newer readers though, Moore's eccentricities can be off-putting. A lot of what he does is riffing off how comics are traditionally done, but that's going to go over your head if you aren't familiar with comics. Watchmen is kind of like the "Citizen Kane" of comics, in that it totally changed the way comics were written. If you want to give Moore a go, a good place to start is his series Top Ten, which presents a world in which everyone is a superhero, and is a whole lot of fun.