Short answers, pulled from the story.
Momentum in Newtonian mechanics is the product of an object's mass and its velocity. It is a vector quantity, meaning it has both magnitude and direction, and its SI unit is the kilogram meter per second, dimensionally equivalent to the newton second.
The law of conservation of momentum states that in a closed system, one that exchanges no matter with its surroundings and feels no external forces, the total momentum stays constant. It follows from Newton's laws and applies to elastic collisions, inelastic collisions, and explosive separations alike.
In an elastic collision no kinetic energy is converted into heat or other forms, as with two highly rigid pool balls. In an inelastic collision some kinetic energy becomes heat or sound, as seen in the damage from a traffic collision, and a bug hitting a windshield is a perfectly inelastic case where both bodies end with the same motion.
John Philoponus introduced an early version around 530 AD, proposing that an impetus is imparted to a thrown object, in contrast to Aristotle's claim that air keeps it moving. The idea was refined by Ibn Sina in 1020 and Jean Buridan around 1350, before Isaac Newton fixed its modern meaning in his Principia in 1687.
Momentum is a measurable quantity whose value depends on the observer's frame of reference. An aircraft of mass one thousand kilograms moving at fifty meters per second has a momentum of fifty thousand kilogram meters per second, but in a five meter per second headwind its momentum relative to the Earth drops to forty-five thousand, and both calculations are equally correct.
Photons carry momentum even though they have no mass, which makes applications such as the solar sail possible. In relativity the energy-momentum relationship holds for massless particles, and in quantum mechanics momentum becomes a self-adjoint operator on the wave function, bound to position by the Heisenberg uncertainty principle.