In layman's terms, a supernova is when a massive star collapses under its own gravitational field. There are many types of supernovae (depending on mass of original star, spectrum etc.). A neutron star is what remains when a Type 2, Type 1b/1c supernova occurs.
These stars consist of mostly only neutrons (0 electric charge, mass more than proton). The mass is somewhere between 1.44 and 3.2 times the mass of the sun. They do not collapse further because of Pauli's exclusion principle (no two fermionic particle can be in the same place and quantum state simultaneously).
Further reading : Chandrashekhar's Limit, Pauli exclusion principle, supernova.
They also have a wonderful infographic of the composition:
As the core of a massive star is compressed during a supernova, and collapses into a neutron star, it retains most of its angular momentum. Since it has only a tiny fraction of its parent's radius (and therefore its moment of inertia is sharply reduced), a neutron star is formed with very high rotation speed, and then gradually slows down. Neutron stars are known to have rotation periods from about 1.4 ms to 30 seconds. The neutron star's density also gives it very high surface gravity, up to 7×10^12 m/s2 with typical values of a few×10^12 m/s2 (that is more than 10^11 times of that of Earth). One measure of such immense gravity is the fact that neutron stars have anescape velocity of around 100,000 km/s, about a third the speed of light. Matter falling onto the surface of a neutron star would be accelerated to tremendous speed by the star's gravity.
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