All right, I saw this movie a couple years ago. It's an older romance movie that was black and white. In it a traveler goes into a town and he falls in love with a girl. After a couple of days, he tells her that he has a finance. She obviously get upset and runs away. The man gets on a train and goes back home. She went to meet the train and stop him, but she misses it and never sees him again.
Years later in New York, the two meet up again. He is married with kids and works for his father-in-law. Their romance lasts though the years although he never leaves his wife. At one point he takes his family and his mistress on a vacation (on a cruise I believe) and one of his sons confronts the woman. At the end of the movie, the man is on his death bed and asks his son to dial a phone number. His mistress answers, he hears her voice, and then dies.

That's all I can remember. Does anyone know the title? Thank you so much, I've been trying to find out what it is for a long time.

I believe you are thinking of the movie Back Street, of which there are three versions, from 1932, 1941, and 1961. The basic story is that some guy meets some woman, but he's engaged, but they have a fling anyway, and then he's willing to run off with her, but due to unexpected events, their rendezvous is disrupted and he leaves town thinking she's blown him off. So he gets married and works for his father-in-law, and years later he meets his crush-girl again, and they resume the affair. Apparently, each movie ends a bit differently. The 1961 version with Susan Hayworth and Vera Miles concludes thusly:

"The situation has become untenable and Saxon decides that the only way to end the pain he has caused Rae is to kill his wife. He chooses a car accident, and of course we understand that the risk will be just a great for him. The wife dies, and Saxon survives – but only long enough to phone Rae and tell her he loves her." And then it seems that she takes in his kids.