Consumer note: the movie is in monochrome. You can't tell if the horse is actually pink. They say it is, though.
Our hero, Gattis, shows up on a Greyhound in the mostly-Spanish speaking town of San Pablo. He mysteriously hides a small piece of paper in a bus station locker, then hides the key (kind of obtrusively). He, for undisclosed reasons, is in search of Frank Hugo, a mobster who's also visiting the town for the local annual fiesta.
Problems ensue: Gattis can't find Hugo right away, neither can he find a hotel room; they're all sold out because of the fiesta. In his search, he accidentally acquires a young hanger-on, Pila, who's (apparently) fascinated by his gringo ways. He also encounters Retz, an FBI agent also on Hugo's case. He eventually makes his way to a local Mexican cantina, where after some initial chilliness, he befriends the clientele, especially Pancho, a fat drunk who runs the local carousel.
On which sits the titular Pink Horse.
It's complicated and nightmarish, and Gattis eventually gets badly hurt. Will he complete his mysterious quest? No spoilers here.
I see from IMDB that the actor playing Pancho nabbed an Oscar nomination for his performance. Losing to Edmund Gwenn, who played Santa in Miracle on 34th Street.