There are nice answers by @fraxinus and @profrob, I would like to add a little side note about the balance between the forces.
It arises because the gravitational field exerted on one body by another is not constant across its parts: the nearest side is attracted more strongly than the farthest side. It is this difference that causes a body to get stretched. Thus, the tidal force is also known as the differential force, as well as a secondary effect of the gravitational field.
Tidal forces are so that they are present when you are freefalling towards a massive body, may that be the Earth, the Sun or a neutron star or a black hole.
Your body is held together by the strong and EM forces (including the covalent bonds), and these forces are dominating over the tidal effects of gravity in most cases of freefall. Do you think that there are no tidal forces in play when free falling here on Earth? Yep there are. It is just that these effects are miniscule and are overwhelmed by the forces that hold your body together.
Only in extreme cases like a black hole (and as you see from fraxinus's answer, only certain black holes) are capable of creating such strong tidal effects, where the gravitational acceleration on your head is so much different from the gravitational acceleration at your feet, that this dominates over the forces holding your body together. In such cases tables turn, the balance turns in favor of the tidal effects and first the chemical bonds are overwhelmed by the tidal effects and your body stretches. But, but in certain cases the effect is so strong, that it dominates over even the EM force holding the electrons and nuclei together, ripping the atoms apart, and eventually dominates over the strong force, thus ripping the the quarks apart.
Even before that happens the gravitational tidal force will rip off the electRons and have the nucleus break up and have most of it converted to neutrons, and then rip those off and get to the quarks, and eventually fall into the singularity. That's why we say that a BH forms when there is too much gravity, nothing can withstand the gravitational effects. Not electron pressure (which holds up white dwarf stars), not nuclear forces (neutron stars), and not strong forces (quark stars, or some parts of the cores of neutron stars).
Yes, the equivalence principle says everything will be accelerated (i.e. pulled) the same way, but only until the force differential between two objects in the atom see different accelerations - that's the gravitational tidal effects, caused by very strong curvatures of the spacetime due to gravity.
Is the gravitational force of Black Hole destroying Atoms?
Now as ProfRob says, here on Earth, when you are in freefall, your whole body can be considered to be in the same inertial frame, because the forces holding your body together dominate over the tidal forces, and these tidal effects can be neglected. In extreme cases, when tables turn and the tidal effects dominate over the forces holding your body together, every single particle in your body needs to be treated as having its own inertial (local) frame that is different from all other particles' frame in your body. "Local" frame in this case will become restricted to a extremely small area (for each particle). Any frame (volume of space) that corresponds to an object that consists of multiple particles cannot be considered to be a "local" frame any more. This is because the gravitational field (and acceleration) varies considerably even on the extremely small scales. I believe this is a very important side note to the answer to your question.