Momentum
Momentum is mass multiplied by velocity, and it points in the direction the object is moving. In any collision the total momentum of everything involved before the impact equals the total afterwards, no matter how messy the collision is. This makes it possible to work out what happens in an interaction without knowing anything about the forces during the interaction itself. In quantum mechanics momentum is still central, but it becomes a property that cannot be pinned down at the same time as position.
Why it matters
It is the cleanest example of a quantity that stays fixed through a process you cannot otherwise calculate, which is why physics reaches for conserved quantities first.
Where it comes up
Read about Momentum on Wikipedia
