This is an extremely good question despite its seemingly simple nature. It connects everything from <PERSON> to string theory to debunking moon landing conspiracies. I'm surprised that the question has '-1' points at the moment (before I upvoted it to make it $0$!).
The concepts surrounding it have a long history involving people from <PERSON> to <PERSON>, <PERSON>, <PERSON>, <PERSON> and even string theorists like <PERSON>. The ideas have been discussed in the context of <PERSON>'s gravity, <PERSON>'s formulation of general relativity, modifications to general relativity and string theory. They have also undergone various, far-reaching, high-precision experimental tests.
<PERSON>'s second law says that a force ($F$) exerted on an object causes it to accelerate ($a$) with the resistance offered to this acceleration being decided by what is called inertial mass ($m_i$):
$$F = m_i a$$
Newtonian gravity says that the gravitational force ($F_g$) experienced by an object depends on the gravitational potential ($\phi$) and its gravitational mass ($m_g$):
$$F_g = -m_g \nabla \phi$$
That $m_i = m_g$, is an empirical fact and/or motivated by deep physical principles. $m_g$ is a measure of how a body reacts to an external gravitational field and is of a different nature than $m_i$. It might very well be the case that $m_i \neq m_g$.
<PERSON>'s empirical observation that pendulums of equal length but different masses have the same time period lead to the idea that all objects are affected by gravity identically. <PERSON> was also instrumental in similar experimental observations, concluding that different bodies undergo free fall identically. These developments lead to the idea that $m_i = m_g$, such that all bodies have the same acceleration under free fall. This idea was that of a universal nature of gravity: that every object (irrespective of its composition/mass) falls at the same acceleration rate in a gravitational field. If $m_i \neq m_g$ then different bodies fall at different rates.
The idea of equating inertial mass with gravitational mass was taken many steps further by <PERSON> with the advent of the Equivalence principle. 766
There are various versions of the equivalence principle, most commonly known ones being the Weak, <PERSON> and Strong equivalence principles: WEP, EEP and SEP, respectively.
1) WEP states that $m_i = m_g$ and therefore, all bodies undergo free fall at the same rate. From the equations of motion, we see that we have a preferred frame, i.e., the free fall frame, where there are no external forces acting on the body. So, locally, being accelerated and being under a gravitational influence have the same physical consequences.
2) EEP states that, in a gravitational field, locally, all non-gravitational laws of physics reduce to that of special relativity.
3) SEP supersedes EEP by including gravitational physics laws in addition to non-gravitational ones.
The <PERSON> experiment is a classical test of the WEP: it checks the equivalence between $m_i$ and $m_g$. The fractional difference in accelerations of different bodies has been measured up to $2 \times 10^{-13}$ according to Ref 1 (page 8).
There is a way to test WEP indirectly through SEP: <PERSON> effect. See also <PERSON>'s original papers: Refs 2 and 3. If SEP is true, then gravitational self energy should contribute equally to inertial mass $m_i$ and gravitational mass $m_g$. If not, then the difference in $m_i$ and $m_g$ causes different objects to accelerate at different rates, violating WEP. Gravitational self energy is negligible for objects used in laboratory experiments (negligible in the sense that such levels of precision are out of the reach of current experimental limits).
But celestial objects have much larger gravitational self-energies. <PERSON> proposed an experiment (see Ref 4) to see if $m_i \neq m_g$, using laser ranging to measure the distance between earth and the moon. This experiment is called the Lunar Laser Ranging experiment. When Apollo 11 astronauts landed on the moon, they placed a retroreflector there to reflect lasers from earth to moon and back to earth for high-precision distance measurements. (Other missions also planted such retroreflectors.) For more information about this experiment, see Refs 5, 6, 7, 8 and 9. 766