Newton's laws of motion and law of gravity very accurately explain and predict natural phenomena, but they are not perfect. There are conditions under which Newton's classical physics does not apply, and one of the areas of modern physics such as quantum mechanics, special relativity, or general relativity apply.
Quantum Mechanics
Quantum mechanics applies in the realm of very small sizes, such as atoms and elementary particles. Newton's laws of physics do not accurately predict the motion or behavior of such small particles, but quantum mechanics does.
For example, wave particle duality, a fundamental principle of quantum physics, states that light waves have particle-like properties and elementary particles have wave-like properties. Through experiments, such as Davisson and Germer's electron diffraction experiment, physicists have observed the wave-like nature of electrons and other elementary particles.
According to quantum physics, macroscopic particles, baseballs for example, also have wave-like properties. However physicists are unable to experimentally observe the wave-like properties of baseballs. As the energy and momentum of particles increases, the wavelength decreases. Subatomic particles, like electrons, have experimentally measurable wavelengths. The wavelengths of baseballs and other macroscopic particles are, however, too small to experimentally measure.
For large particles quantum mechanics is unnecessary. Newton's physics works well.
Special Relativity
Einstein's special relativity theory predicts many effects, such as time dilation and Lorentz contraction, that violate common sense. Special relativity effects only become easily measurable at speeds greater than about 10% of the speed of light. Because people never travel this fast, special relativity seems counterintuitive.
Elementary particle physicists do however frequently accelerate elementary particles to speeds near the speed of light. At such high speeds Newton's laws do not work. Physicists must use special relativity to predict how particles will behave.
Newton's physics works at low but not high speeds.
General Relativity
Einstein's general relativity theory explains gravity. Newton's law of gravity describes gravitational effects but does not explain why gravitational forces occur. General relativity theory suggests that masses curve four dimensional space-time to produce gravitational forces.
For relatively weak gravitational fields, such as on Earth, Newton's gravity and Einstein's general relativity are indistinguishable. For strong gravitational fields, found very close to the Sun or a black hole, Newton's law of gravity gives incorrect results. General relativity however agrees with experimental data.
For example, the perihelion (point in the orbit closest to the Sun) of Mercury, migrates a little as Mercury orbits the Sun. The amount of migration, called precession of the perihelion, is correctly predicted by general relativity but not by Newton's physics. The gravitational force is large enough that Newtonian gravity is less accurate.
Newton's Physics, Modern Physics, and the Scientific Method
These limitations to Newton's physics provide a good example of how the scientific method works. Rather than proving theories, scientists test and fail to disprove theories. If experimental data cannot disprove a theory, scientists tentatively accept the theory as correct. If experimental data disagree with a theory, scientists discard or modify the theory. All scientific theories are subject to change as dictated by experimental data.
Scientists accepted Newton's classical physics because it agreed with experimental data and was not proven wrong.
By the late 19th and early 20th centuries, however, scientists found experimental data that disagreed with Newton's classical physics under certain conditions. Special relativity, general relativity, and quantum mechanics are modifications to Newton's physics for conditions where classical physics is inaccurate. When the conditions do not require these modifications, Newton's physics is a sufficiently accurate approximation.
Relativity and quantum mechanics are probably not the ultimate physics theories. Someday physicists will discover conditions under which modern physics does not quite work. Physicists will then modify relativity and quantum mechanics, just as they modified Newton's classical physics in the early 20th century.