Policeman Rick Grimes is shot attempting to apprehend an escaped convict, but is himself shot in the attempt. He falls into a coma, and when he awakens, he finds a changed world. There is no government, no police, no society at all. Just a small number of survivors. And lots and lots of zombies.
I have not seen an episode of the TV show based on this comic, and this is my first time reading the issues. As many people have said before, these are excellent comics -- the story is fast-paced, the characterizations are believable, and the dialogue is realistic. These are the types of comics that demonstrate that sequential storytelling is a medium, not a genre. Any type of story can be told in this format.
As I mentioned when I reviewed a Showcase reprint collection, comics that are designed to be in color, yet shown in black-and-white can be hard to look at. But this comic is designed to be viewed in black-and-white, and the artists make great use of this color choice. Moore & Adlard do a great job of differentiating the looks of the large cast of characters, an impressive feat without being able to show color differences in hair or clothing.
These issues are collected in the trade paperback collections "Days Gone By" (issues 1-6) and "Miles Behind Us" (7-12), and also in the Hardcover collection, volume 1 (all 12 issues).