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.