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According to 14 CFR 61.109, a private pilot applicant must log 40 hours of flight time.
(...) that includes at least 20 hours of flight training from an authorized instructor and 10 hours of solo flight training (...)
What are the requirements for the remaining 10 hours?
There is no requirement for the remaining 10 hours, just that those 10 hours are part of your instruction/training.
So really you can read it as "a private pilot applicant must log at least 40 hours, of which 20 must be flight training from an authorized instructor and 10 hours must be solo".
How you distribute the remaining 10 are up to you and your instructor, although the vast majority of students don't find it difficult to place those hours anyway since the national average is closer to 70 hours to get a PPL.
There are also specific requirements for logging time doing certain things like solo cross-country flight, night flights, or landing at airports with a tower. From the Wikipedia article on US Pilot Certification.
Accumulate and log a specified amount of training and experience, including the following: If training under Part 61, Title 14 of the Code of Federal Regulations (CFR) section 61.109, requires at least 40 hours of flight time, including 20 hours of flight with an instructor and 10 hours of solo flight (i.e., by yourself), and other requirements including cross-country flight, which include:
Solo requirements:
5 hours of solo cross-country time
One solo cross-country flight of at least 150 nmi (280 km) total distance, with full-stop landings at a minimum of three points and with one segment of the flight consisting of a straight-line distance of at least 50 nmi (93 km) between the takeoff and landing locations.
Three solo takeoffs and landings to a full stop at an airport with an operating control tower.
Night requirements:
3 hours of night flight training
10 takeoffs and 10 landings to a full stop (with each landing involving a flight in the traffic pattern) at an airport
1 cross country flight of 100 nm total distance at night
3 hours of flight training on the control and maneuvering solely by reference to instruments
3 hours of flight training for cross country flights
3 hours of flight training with an authorized instructor in preparation for the practical test, which must have been performed within the preceding 2 calendar months from the month of the test
