The speaker is Holden Caulfield, the narrator of the cult novel "The Catcher in the Rye", by recluse writer J.D. Salinger. Holden is a teenager who escapes a boarding school in order to spend a few days in New York, where he interacts with strangers and experiences new things.
Meaning and context: When Holden says he has Jane Gallagher on the brain again, he means he cannot stop thinking about her. Jane is a girl whom he deeply admires, but at the same time he never makes the first move. When he learns his roommate has a date with Jane, he is assaulted by jealousy. The complete quote goes like this:
"All of a sudden, on my way out to the lobby, I got old Jane Gallagher on the brain again. I got her on, and I couldn't get her off."