There are actually two Chandrasekhar Limits, and both are limits on the mass of a stellar remnant after a catastrophic event in the star's life, carbon flash, has occurred.
When a star's core is smaller than about 1.44 solar masses, the remnant's electron degeneracy pressure is enough to counteract gravity, and the remnant becomes a white dwarf made up of degenerate matter. Stars with an original mass of up to 5 solar masses can blow away enough material to stay white dwarfs.
This limit was actually calculated by Subrahmanyan Chandrasekhar in 1930 when he was 20 years old, on the ship from India to Great Britain to study at Cambridge University.
When a star's core is greater than 1.44 solar masses, the equations for electiron degeneracy pressure versus gravity have no solution. This means the remnant can no longer stop further collapse, and there is a runaway series of nuclear fusion reactions culminating in a Type II Supernova. Electrons are pushed into atomic nuclei, turning the remnant into a neutron star, essentially a big mass of neutrons (neutronium). If an existing white dwarf can gain enough mass (say by stealing from a companion) to send it over this limit, it will also collapse into a neutron star. When the core is less than about 2 solar masses1, the neutrons are able to resist further collapse through a QCD analogue of the Pauli Exclusion Principle. This limit was probably calculated by Karl Schwarzschild but it is often referred to as "Chandrasekhar's Upper Limit".
When a star's core is greater than 2 solar masses, the neutrons can no longer halt collapse and the remnant becomes a black hole. If an existing neutron star can gain enough mass to send it over this limit, it will also collapse into a black hole.
On a completely unrelated subject, "Chandrasekhar's Other Limit" is a catch phrase within Frederik Pohl's 1982 novel Starburst. Pohl relates (via his half-dead, half-alive character Will Becklund, also known as "Uncle Ghost") a story Chandrasekhar allegedly told about dragonfly larvae who break through the surface of the water as they undergo their final metamorphosis into adults. There is no possibility for communication between the two states. The concept refers, of course, to death, or perhaps the afterlife, if there is one.
1unperson informs me the actual value of the upper limit is a matter of some debate.