Night of the Living Dead (1968)
Night of the Living Dead was a groundbreaking film when it first came out and it was the 1st in a long list of George A. Romero zombie flicks in the 'dead' series. Shot in black and white NOTLD is basically the second zombie movie ever made (the first being White Zombie).
The film starts off with a woman named Barbara (Judith O'Dea) and her brother at a cemetery visiting their father's grave. Out of nowhere, the undead begin chasing them (in slow motion like the old school Mummy movies). Barbara makes it out alive and into a house that begins to fill up with random other survivors anchored by the movies hero, Ben (Duane Jones) who is a black man. I only mention this because, at the time, this was atypical of movie heroes. Kudos to Romero for not being a dumb racist jackass.
This movie is not for you if you want fast-paced zombies with super human strength, BUT if you want to see where the phenomenon started, then this is the movie for you.
Dawn of the Dead (Remake-2004)
Keeping in line with the 'Dead' series I am going to highlight Dawn of the Dead. BUT I am going to recommend the remake. Though the original is fantastic, the remake was done very well and shot in a very different way from the original. The opening scene alone makes the DOTD remake one of the best horror movies this decade.
Unlike zombie movies of the past, this remake offers a fast-paced, strong, and fairly intelligent undead population. Don't worry, they are still out looking for some good old fashion brains to chow down on.
There is a well known cast of actors attached to this, none outshining the fairly unknown (at the time) Ty Burrell who is simply awesome and is, incidentally, in one of my new favorite TV shows