A core curriculum is a set of courses that must be taken by all students. The Chicago core curriculum includes 2-3 course humanities sequence, 3 course social sciences sequence, 2-3 course civilizations sequence, 1-2 courses in the fine arts, 1-2 courses in mathematics (usually 2 courses calculus), 2-3 courses in the physical sciences, 2-3 courses in the biological sciences, and the passing of a language competency exam, usually equal to one year of study at the college level. The hum, sosc, and civ sequences all combine a variety of academic disciplines. For example, sosc includes anthropology, philosophy, history, psychology, and probably some others. Classes are small and discussion-based with a lot of faculty interaction.
You will get a great education at either school, without a doubt. The cores have a few differences, and the atmospheres are a little different. I don't know the Columbia Core off the top of my head, but I know that at Columbia there are certain classes that every student takes. At Chicago, the core includes choices so that the student can pick a particular lens or focus for their study. For humanities at Chicago, you could take Philosophical Perspectives on the Humanities, Human Being and Citizen, Readings in World Literature, Reading Cultures, Media Aesthetics, and many another one or two that I'm forgetting. At Columbia, each student takes the exact same classes.
You can read more about Chicago's requirements here: http://collegecatalog.uchicago.edu/pdf_08/Curr.pdf
And this is Chicago's course guide: http://collegecatalog.uchicago.edu/
You should try to visit each campus if possible and go on a campus tour, talk to students, etc. The University of Chicago is particularly known for having interesting, creative, and intellectually-driven students. Columbia probably has a bit of a different feel. Read about both schools on their websites and visit. Both schools are fantastic. It would be stupid to say that you would get a better education at one or the other.