What does DI mean?

DI stands for Digital Intermediate.

A digital intermediate "session" is what takes place when a movie and or TV commercial is 

in the very final stages of completion.

What usually happens is the DP ( Director of Photography, Cinematographer, Cameraman) sits in a very dark room with a person that is called a Colorist who works on really cool looking machines that cost lots of money and make corrections to the color of a shot or particular piece of a shot or make a color movie a black and white movie.

This is all done digitally, ( as opposed to Photo Chemically) which is then stored on hard drive and sent to a Arri Film Scanner where the corrected information is then laser scanned onto FILM Negative so it can then be sent

to a place called Deluxe Laboratories so they can make 1000's of FILM Prints for the movie theaters.