Short answers, pulled from the story.
Quantum mechanics is the fundamental physical theory that describes the behavior of matter and of light, with its unusual characteristics appearing at and below the scale of atoms. It is the foundation of all quantum physics, including quantum chemistry, quantum field theory, and quantum information science. Classical mechanics can be derived from it as an approximation valid at ordinary scales.
Quantum mechanics usually cannot predict with certainty what will happen and only gives probabilities. A probability is found using the Born rule, named after Max Born, which takes the square of the absolute value of a probability amplitude. This probabilistic nature stems from the act of measurement.
The uncertainty principle states that no preparation of a quantum particle can give precise predictions for both its position and its momentum at the same time. Position and momentum are represented by Hermitian operators that do not commute. Either standard deviation can be made arbitrarily small on its own, but not both simultaneously.
Quantum mechanics arose gradually, beginning with Max Planck's solution to the black-body radiation problem in 1900 and Albert Einstein's 1905 explanation of the photoelectric effect. The full theory was developed in the mid-1920s by Niels Bohr, Erwin Schrodinger, Werner Heisenberg, Max Born, Paul Dirac, and others. By 1930 it was further unified and formalized by David Hilbert, Paul Dirac, and John von Neumann.
Quantum entanglement is when the properties of interacting quantum systems become so intertwined that the whole can no longer be described in terms of its individual parts. Erwin Schrodinger called it the characteristic trait of quantum mechanics. It cannot send signals faster than light, as shown by the no-communication theorem, but it does enable quantum computing and protocols such as quantum key distribution.
Quantum mechanics underpins solid-state physics and materials science and explains the behavior of individual subatomic particles. Its applications include quantum chemistry, quantum optics, quantum computing, superconducting magnets, light-emitting diodes, the laser, the transistor, and semiconductors such as the microprocessor, along with imaging methods like magnetic resonance imaging and electron microscopy.