What are the requirements to get your private pilot certificate?
In the US, you'll need to be 16 old to start training and 17 to get your Private Pilot's license in a piston-powered aircraft. To get your license you'll have to have a minimum of 40 hours total time, including at least 20 with an instructor and 10 solo. You will also have to pass a written test, and get a Medical certificate.

Details: 

If you're pursuing a Private Pilot license, per 14 CFR §61.83(a) or (b) you must be 16 years old to get a student pilot certificate (or 14 years old if you're going to train in a balloon or glider.)   

NOTE: 14 CFR mean Code of Federal Regulations, Title 14 - Aeronautics &amp; Space.  §61.## means "Part 61" which covers Certification of Pilots and Instructors.

To start training you will also have to pass a medical exam for a 3rd Class Medical Certificate, per 14 CFR §61.23(a)(3). If you're generally healthy, getting a 3rd Class Medical isn't terribly difficult.

Assuming you successfully complete the training, per 14 CFR §61.103(a) or (b) you can get your Private Pilot license for an airplane at 17 years old (or 16 years old for a balloon or glider.)

Per 14 CFR §61.109(a) to get your license you will need to have completed at least 40 hours of flight time, including at least 20 hours of time with an instructor and 10 hours of solo time. 

However, many pilots take more than the required minimum 40 hours to get their license. My suggestion is to try to complete your training in as short a calendar time as is possible. By taking lessons frequently (more than once per week) you will minimize the amount of "repeat" lessons, which will help you get your license with fewer hours. If you're only taking a lesson every few weeks, you will end up repeating topics and spending more than the required 40 hour minimum.

Per 14 CFR §61.35, you will have to complete a ground school program (can be with an instructor, or a "home study" course) before you take the Knowledge test (aka: written exam.)

Per 14 CFR §61.109(a)(3), you will need at least 3 hours of "Instrument" flight training to get your Private Pilot license. That's really not intended to be enough for you to fly safely on instruments, but to give you a basic set of skills to get out of a low-visibility situation. If you keep up with your flying, getting in Instrument rating is a GREAT idea, once you have enough hours.

You can look up the specific wording of all of the rules at http://www.ecfr.gov/

Another good resource for student pilots is the Aircraft Owner's and Pilot's Association's Learn to Fly page http://www.aopa.org/Pilot-Resources/Learn-to-Fly.aspx.