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Neutron stars are the remnants of massive stars that have undergone a core-collapse supernova. At the end of the star's life, when the nuclear fuel has been exhausted, the core becomes unable to hold out against gravitational forces. The collapse is halted by neutron degeneracy pressure, which is a quantum pressure arising from the principles laid out by the Pauli exclusion principle. This prevents the neutrons from occupying the same space, resisting further compression.

The result is an incredibly dense object primarily composed of neutrons. A neutron star's mass typically ranges between 1.4 to 2 solar masses but spans only a city-sized diameter of about 20 kilometers. The density of neutron stars is so immense that a sugar-cube-sized amount would weigh roughly as much as humanity combined. Such stars often exhibit rapid rotation and strong magnetic fields, sometimes observed as pulsars, due to their regular emission of radiation.

Black holes are formed when massive stars end their life cycles with a core-collapse supernova, but under more extreme conditions than neutron stars. When the mass of the stellar core exceeds a critical threshold after the explosion, not even neutron degeneracy pressure can prevent the collapse. This threshold is known to surpass the Tolman-Oppenheimer-Volkoff (TOV) limit.

As gravity triumphs over all other forces, the core continues collapsing to a point, or singularity, where density and gravity become infinite. The gravitational pull of a black hole is so intense that nothing, not even light, can escape, making them invisible to direct observation. Black holes can be detected through their influence on nearby stars and the light from the material accreting around them. Their formation often leaves an observational signature, such as high-energy emissions from the surrounding accretion disk.

The Tolman-Oppenheimer-Volkoff limit, named after scientists Richard Tolman, J. Robert Oppenheimer, and George Volkoff, establishes the maximum mass a neutron star can have before collapsing under its own gravity to form a black hole. The TOV limit is parallel to the Chandrasekhar limit, which pertains to white dwarfs.

Typically estimated between 2 to 3 solar masses, this limit is a cornerstone in astrophysical research, though it still remains a subject for further study due to uncertainties in the composition of neutron stars and the equation of state, which describes how matter behaves at such high densities.

Understanding the exact TOV limit is crucial in astrophysics as it helps predict the fate of collapsing stars and guides the characterization of exotic objects formed in the cosmos. It remains a fascinating example of how theoretical physics and observational astronomy work hand-in-hand to unravel the mysteries of the universe.