Asano Inio writes about people. That is to say, he does not write about magic Asian wizards or the deeds of men, but he writes about real people. Real peoples' lives aren't very exciting. Of course, that doesn't mean it's easy either. And the worst part? Everyone else is going through the same thing as well.
Asano Inio understands this, though, which is why he's able to write some compelling stories about the melancholy of your average person living life aimlessly. Of course, since he's a manga-ka, he's not all talk either. He's got a distinct style to his characters, which are instantly recognizable, but he also has some amazingly detailed spreads and backgrounds.
So what has he done?
We start off with What a Wonderful World, his first work. It's a series of short stories with the characters and events extremely loosely connected. It involves people of all sorts: kids, teens, college students, adults. My favourite little detail about WaWW is the transitional panels in between each chapter. But yes, there is an overarching theme that you may need a second reading to get the finer details of.
After that, we have Nijigahara Holograph, a psychological horror story with a strange fixation on butterflies. Just like WaWW, there are a few plot threads, but the connections are a lot deeper and heavier in this story. This one is a lot darker; you'd probably want to be a little older to read this. It's definitely Higurashi-class horror. This one, you'll definitely need a few readings to take it in and understand.
Solanin is probably his most famous work. It's also his most accessible. While the previous two series had some strange supernatural stuff thrown in, Solanin is fairly down to earth. It's about some recent graduates who sort of fell into jobs and now aren't sure what to do with themselves and the dreams they had when they were still students. This one hit me hard like 5 Centimeters per Second.
Oyasumi Punpun is his most recent and longest work. It's about a regular boy, Punpun. Except that he and his immediate family appear to us, the readers, as a caricaturized bird thing. Otherwise, he acts and thinks exactly like a normal kid would. Oyasumi Punpun is about his growing up. I can't mention enough how brilliant the fact that Punpun is abstracted like this is. Believe me when I say that it doesn't take any emotional impact away from his circumstances at all.
I don't know if you've noticed, but Asano Inio is my favourite manga-ka. Besides his art being absolutely gorgeous, he has a knack for writing realistic characters and conveying their emotions. Just like real life, his stuff runs the gamut, from adorable or amusing to cripplingly depressing. And on both ends of the spectrum, the scary thing is that all of these stories are completely possible and believable.
If you're new to his works, I'd definitely recommend going through Solanin first and then WaWW, since they're waaaaay more accessible than his other stuff. After that, you should have enough of a grasp on his style to be able to go through his other (just as awesome) works. Also, Solanin and What a Wonderful World are the only works of his that are available in English for purchase.