The private pilot certificate is the one held by the majority of active pilots. It lets you fly just about any aircraft (subject to appropriate ratings) for any non-commercial purpose, and gives almost unlimited authority to fly under visual flight rules (VFR).
Private pilots can carry multiple passengers, unlike sport pilots who can carry only one, and private pilots can fly at night and in more types of airspace with no distance restrictions as compared to sport pilots. With more training and an additional rating, private pilots can also operate under Instrument Flight Rules.
When pursuing a Private Pilot certificate, you can choose between Part 141 and Part 61 curricula, so named for the section of the FAA regulations that govern them. Part 141 training is more formalized and is often found as part of a larger flight school, while Part 61 usually represents a more individualized one-on-one training program between you and your flight instructor.
The minimum required training time for a private pilot certificate is 35 hours for Part 141, and 40 hours for Part 61.