The information here refers specifically to applications sent by mail. It is also the recommended deadline, since some departments begin reviewing files and making decisions in early December. The reduced fee applies only to fall applicants. All students seeking fall admission should meet the December 31 deadline, since it is likely that enrollment in many departments will be filled soon after this date. Late applications may be considered for admission only if all spaces have not been filled, and for financial aid only if funds are still available.
Departments vary in their willingness to consider late applications.
Other departments will continue to review applications until all spaces have been filled. You should read carefully all the information on deadlines in this website. In every case, the earlier you get materials to the Graduate School, the better your chances for full consideration. However, very few departments actually continue to review applications this late. Note that all financial aid funds have usually been committed well before the summer.