If you are flying as a hobbyist
If you are flying as a hobbyist, which seems to be the case, you are required by Public Law 112-95 FAA Modernization and Reform Act of 2012 section 336 to:
(5) when flown within 5 miles of an airport, the operator of the aircraft provides the airport operator and the airport air traffic control tower (when an air traffic facility is located at the airport) with prior notice of the operation (model aircraft operators flying from a permanent location within 5 miles of an airport should establish a mutually-agreed upon operating procedure with the airport operator and the airport air traffic control tower (when an air traffic facility is located at the airport)).
This, of course, does not say how you are to contact the airport operator and air traffic control. The traditional way would be to find the telephone numbers for the airport and air traffic control, and to call them. Alternatively, as you discovered, some mobile phone apps allow you to electronically submit notices to some airports. What is not required or desired is to contact ATC by radio. If you are operating very near an airport, it may be a good idea to get a radio so that you can listen in.
If you are flying under Part 107
If you are flying under Part 107, you do not have to contact the airport or air traffic control even if you are within 5 miles of an airport, but you must not fly in controlled airspace without obtaining an airspace authorization or airspace waiver. These processes are, at this time, handled through the FAA Drone Zone website, and not by contacting the airport or ATC. Depending on the terms of the authorization or waiver, if granted, you may be required to contact ATC prior to operation, and they will need a way to get in touch with you.
Which rules do you need to follow?
Broadly, if you're flying for fun (e.g. as part of an enthusiast group, in your back yard, or taking personal photos in the park (although watch out for people as per the safety guidelines you are following)), you are a hobbyist, and subject to the special rule for model aircraft.
If you're flying for commercial or business purposes (e.g. taking photos of houses for real estate sales, inspecting railroad tracks, or delivering purchased packages), you need a Part 107 license and to fly under Part 107 rules.
For more information, see the FAA's sUAS FAQs and the table on page 11 of the FAA's interpretation of the Special Rule for Model Aircraft (pdf).