The Red and the Black refers to the color that people in 19th-century France would wear to show that they had certain types of jobs. It was usually soldiers who wore red and priests who wore black. At the beginning of this book, Julien wants to climb the social ladder by making a name for himself in the army (the red).
Unfortunately, it was no longer possible to do this in France, which is why Julien is nostalgic for the good old days of Napoleon Bonaparte, when even the lowliest peasant could distinguish himself in war. Julien eventually turns to the priesthood (black) in order to fulfill his social ambitions. The only problem is that he doesn't really believe in any of the religious stuff he's supposed to preach.
Another possible meaning for red and black in this book would be love (red) and death (black). If you recall, Julien forms romantic connections with two women in this book, and these passionate connections eventually lead to his death… which would symbolically show a transition from red into black.