I now understand that weight causes pressure, as does temperature, as does anything else that applies a force. Is this correct?
Yes, I think you have it. There are more ways to describe pressure (e.g. particle density may be a factor), but everything you said jives well with me.
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Now let me mess with your head.
Hydraulic Pressure (what we've been discussing) is a statistical averaging of a huge number of miniscule forces over an area.
Hydraulic pressure is mostly electromagnetic repulsion of the electrons in the outermost shells of atoms and molecules. (It is also the resistance to bending of the inter-molecular bonds - still electromagnetism.) From a "distance" an atom has neutral charge because it has an equal number of protons in the nucleus and electrons in the electron shells. Since protons and electrons have equal and opposite charges, they cancel out, with a net charge of 0. However, "up close" the electrons are on the outside, shielding the positive charge on the inside. This means that the electrons of one atom are close to the electrons of the other atom... and they repel, as like charges do, by expressing the Coulomb Force on each other.
The Coulomb force is the electromagnetic force that says, "Like charges repel. Unlike charges attract. Here's how much."
It is stylistically no different from Newton's Law of Gravitation.
Spoiler:
Newton:
F = -GMm/r^2
Gravitational force, F, equals Newton's Gravitational Constant, G, times the mass of one body, M, times the mass of the other body, m, divided by the square of the distance between them, r^2. The minus sign tells us that like "mass-charges" attract, and unlike "mass-charges" repel. Since all mass has positive "mass-charge", the force is always attractive. (There is no observation of negative mass.)
Coulomb:
F = {pink elephant} Qq/r^2
Electromagnetic force, F, equals {pink elephant} times the charge of one particle, Q, times the charge of the other particle, q, divided by the square of the distance between them, r^2. The lack of minus sign means that like charges repel, and unlike charges attract.
{pink elephant} = 1/(4*pi*{ongbonga})
where {ongbonga} is the electrical permittivity of free space, usually {epsilon_0}
There are stronger forces, which can result in pressures under "exotic" conditions.
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Totally different kinds of pressure (but still the result of QM forces on particles):
When a massive star begins fusing Iron, its core will collapse, and it will supernova. This is because Iron takes more energy to fuse than it gives off in having fused. It is the lightest element to have this property. Once the star begins to fuse Iron, the pressure in the center goes up and it fuses Iron faster, which makes the pressure go up... it's the chain reaction that is the death-knell of the star.
(A star had to die for there to be Iron to make blood. *mind blown every time*)
How does a thing which has stopped pushing outward against the gravitational force suddenly explode against that force?
The core collapses, falling to the center. eventually the electrons are pressed so close together that QM basically just says "No." This is the electron degeneracy pressure. The electrons are facing the limit of particle density described by the Pauli Exclusion Principle. The PEP says that no 2 fermions can be in the same state at the same time. Electrons are fermions. When the core collapses, all of the particles rush to the center of the star, which quickly fills up. Once those electrons are at maximum particle density, then you have something which makes "incompressible water" look like a sponge.
That electron degeneracy pressure is so great that the in-falling star bounces off of it at nearly the speed of light. Supernova
BUT, that's not the "hardest" pressure.
If the star is big enough to leave a stellar remnant, The result could be as "exotic" as a neutron star.
In a neutron star, the electron degeneracy pressure was overwhelmed. The PEP would not yield, and so the electrons could no longer exist in the same volume and the increased pressure of the in-falling star forced the electrons to bond with the protons into neutrons.
A neutron star has the greatest mass-density of anything that is not a black hole. The entire object is almost wholly composed of neutrons, with only a small percentage of protons and electrons that are unbound. The pressure is too great for the electrons and protons to separate and form atoms in any notable amount.