CubMail is Columbia's webmail program. The name is a reference to a baby Columbia Lion, in case you couldn't figure that out.
A "new" and "improved" version was released in 2008, but it's really just the old version with slightly darker interface colors. As of 7 July 2009, the old version is no longer available.
A better alternative is to use Gmail to send and receive email from your Columbia.edu address.