Coordinate system
Definition
The grid and rules you lay over space to say where things are — like choosing where 'zero' sits and which way is positive before solving any motion problem.
Technical definition
A reference framework assigning ordered numbers (coordinates) to each point in space: Cartesian (x, y, z), polar (r, θ), and others. Physics results are independent of the choice, but a smart choice of origin and axes simplifies the mathematics enormously.
Example
For a projectile problem, placing the origin at the launch point with +y upward turns the motion into x = v₀ₓt and y = v₀ᵧt − ½gt². The same problem with the origin at the landing point would need constant offsets everywhere — same physics, messier algebra.
There is no 'correct' coordinate system handed down by nature — any consistent choice works, and signs follow your choice, not the object's. A ball falling downward has positive velocity if you chose +y downward. Most sign errors come from mixing two different choices mid-problem.