The first full feature length zombie movie was White Zombie in 1932. It was produced and directed by the Halperin brothers and starred the famous horror actor Bela Lugosi as an evil voodoo practitioner who 'kills' an innocent young woman and turns her into a zombie. A sequel was made in 1936, called Revolt of the Zombies.
Compared to most later zombie films, White Zombie is quite tame, and is even rather romantic (I won't give away the plot, but the young woman is given a 'potion' in the hope that she will fall in love...). Critics didn't think much of it at the time, but it was quite successful and definitely had long lasting effects in creating the zombie genre.
This is the first real zombie movie, but there were earlier examples of films involving bringing the dead back to life or similar themes, most famously the first Frankenstein film which was released in 1910.