A "Student Pilot" certificate is just what it sounds like; someone who has no aviation experience and is first learning to fly. A person is officially a student pilot after a medical exam and when they receive a Medical Certificate from an approved examiner (see list of examiners).
You can start flying with an instructor before receiving the Medical Certificate, but must have a Medical Certificate before you can fly solo (fly by yourself), which happens somewhere after 10 to 15 hours of dual instruction (flying with an instructor in the airplane).
A student will fly with an instructor typically from 20 to 30 hours and the rest of the time they will fly by themselves. You'll realize quickly that keeping up with flight time is very important (it's call logging your flight time). Flying with an instructor is called dual-instruction and flying by yourself is called solo. There is also 10 - 20 hours of ground training from an instructor.
After logging at least 40 hours of flight time, passing a written test (see list of places to take written test), and passing a practical exam, you will earn a Private Pilots license.
A "Private Pilot" can fly themselves, family or friends just about anywhere (except for very special airspace, like military bases) and into any public airport, big or small. Yes, you could legally fly into Atlanta or LaGuardia.
A Private Pilot can flight a variety of aircraft, depending upon the type of aircraft they trained in when they were a Student Pilot or types of aircraft they received special training in after obtaining a Private Pilots license. For Aileron Aviation students this means light, single engine airplanes, like Aileron Aviation's Cessna 172.
Bigger or more complicated aircraft will require additional training. For example, airplanes with retractable gears or airplanes with multiple engines. The cost and effort to obtain a qualification for these types of aircraft is smaller than the cost and effort to obtain an initial Private Pilot license.
A "Commercial Pilot" can be paid to fly. This isn't the same as flying for business trips. A private pilot can make personal business trips. A commercial pilot can be paid to be a pilot. This is a necessary certificate for anyone pursuing an aviation career, but it's also a great accomplishment for every pilot. A commercial pilot develops mastery of the aircraft.
An "Instrument Rating" allows a pilot to fly in the clouds, rain or in any other low visibility situation. Pilots use the airplane's instruments to maintain level flight, climb, descend and turn. An instrument pilot develops a high degree of confidence and ability to communicate with Air Traffic Control ("ATC").