Yes, if you are overseas and planning to study in the ELA Intensive English Program, you will need an F-1 student visa.
Using the SEVIS ID number from your I-20, you will make an appointment with the U.S. Consulate in your country and take these documents with you to apply for your visa. You will also need to pay a SEVIS I-901 fee, which you can do online. Your I-20 is for a specific start date when you must report to the ELA to begin your program. If you delay or defer your enrollment, you will need a new DePaul I-20.
More about information about visas can be found here.
US Citizens, Permanent Residents (Green Card Holders), holders of Religious Worker Visas and certain other visa types may study at ELA without needing an I-20.