Short answers, pulled from the story.
Angular momentum is the rotational analog of linear momentum, measuring the quantity of rotational motion an object or system possesses. It is conserved in isolated systems because of rotational invariance: whenever the laws of physics remain unchanged under rotation, Noether's theorem guarantees conservation. No external torque means no change in total angular momentum.
Bicycles, motorcycles, flying discs, rifled bullets, and gyroscopes all rely on conservation of angular momentum for their stability. Hurricanes spiral because air molecules accelerate as they are drawn inward toward a low-pressure center, conserving angular momentum. Neutron stars spin extremely fast because a large, slow-rotating star collapses to a tiny radius, dramatically decreasing its moment of inertia.
Tidal torque from the Moon transfers angular momentum from Earth's spin to the Moon's orbit. Earth's rotation slows by about 65.7 nanoseconds per day, while the Moon's orbital radius increases by about 3.82 centimeters per year. The total angular momentum of the system remains constant.
William J. M. Rankine defined angular momentum in the modern sense in his 1858 Manual of Applied Mechanics, describing it as a directed line whose length is proportional to the magnitude and whose direction is perpendicular to the plane of motion. The term itself appears in earlier publications from the late eighteenth and early nineteenth centuries, predating Rankine's attribution to R. B. Hayward.
In classical mechanics, angular momentum can take any continuous value and has a well-defined direction. In quantum mechanics, angular momentum is quantized, jumping between discrete values set by the reduced Planck constant (approximately ten to the power of negative thirty-four joule-seconds). Only one component of the angular momentum vector can be measured precisely at any time; the others remain uncertain by the Heisenberg uncertainty principle.
Spin angular momentum is an intrinsic property of elementary particles, distinct from the angular momentum arising from motion through space. Electrons have spin one-half, photons have spin one, and pi-mesons have spin zero. The Higgs boson is the only known elementary scalar boson with spin zero. Spin does not correspond to any physical spinning motion.