I have recently done some research into black holes, and realized that big stars form black holes, whilst smaller ones don't. Is this because the gravity isn't strong enough for it to fall in on itself, or something else?
Gravitational collapse occurs when an object's internal pressure is insufficient to resist the object's own gravity. In the cases of stars, it normally usually happens because of one of these 2 reasons:
The star has too little "fuel" left to maintain its temperature
The star that would have been stable receives extra matter in a way that does not raise its core temperature.
In either of these cases, the star's temperature is no longer high enough to prevent it from collapsing under its own weight (gravity). The collapse may be stopped by various factors condensing the matter into a denser state. The result is one of the various types of compact star.
As said above, it needs a lot of mass to actually collapse into either a black hole, or implode as a supernova, and smaller stars simply don't have this.
The actual answer has nothing at all to do with the temperature.
Even low-mass stars would form black holes if they ran out of nuclear fuel to burn, and simply cooled whilst being supported by "standard" gas pressure in their centres.
That is because that gas pressure would be proportional to the temperature, but the star is able to cool so it would need to shrink and use gravitational energy to stay hot, but of course eventually it would disappear inside its event horizon and become a black hole.
The real reason this does not occur is electron degeneracy pressure. This is a quantum mechanical effect, related to the Pauli exclusion principle, that does not allow two electrons to occupy the same quantum state. As the gas gets squashed in the core, the electrons are forced to fill higher and higher momentum states in order to "avoid" each other. Because these electrons have large momentum they also exert a large pressure - degeneracy pressure. This degeneracy pressure does not depend on temperature and so the star (it is a white dwarf when a low-mass star arrives at this stage) can cool and remain in hydrostatic equilibrium without shrinking any further. Thus it avoids becoming a black hole (or even a neutron star) and simply cools at nearly constant radius.