Please see the CS department's webpage on admissions. There, you'll find information about all aspects of the application process (e.g., deadlines, your chances of being admitted), as well as information about the academic programs themselves.
If you are interested in working with me, you can mention this in your application (and also see the section below); this will draw my attention (as well as others') to your file. Unfortunately, I do not generally have time to reply to inquiries from prospective students.
I work on algorithmic statistics and machine learning, with emphases on probabilistic models (especially latent variable models), interactive learning (e.g., active learning, online learning), and privacy-preserving data analysis. My recent papers can provide some examples of these topics.
If you are a current (or already admitted) Columbia student, please feel free to contact me via e-mail (or drop by my office) if you are interested in discussing a project with me.