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A set of mathematical axes which serve as a quantitative map grid, allowing precise specification of positions of objects. Cartesian coordinates are most common in introductory mechanics, but cylindrical coordinates are sometimes useful, especially for circular or orbital motion.

Setting Up Coordinate Systems

Problems Involving Motion

Sketch an x-axis (and, if needed, a y-axis).
Clearly specify which direction is positive.
Clearly specify where you are choosing to place the origin.

Problems Involving Dynamics or Momentum

Only the orientation of the x- and y- (if relevant) axes need be shown. The precise origin of the axes is usually not important.

Problems Involving Energy

For problems involving (near-earth) gravitational potential energy it is assumed that up is the positive direction for height. You must, however, specify a zero-point for the height.
For problems involving springs, it will be assumed the origin is placed at the equilibrium position of the spring, unless otherwise specified.
For problems involving (near-earth) gravitational potential energy and springs, you must clearly describe the relationship between the coordinate used in the gravitational potential energy and the coordinate used in the spring potential energy.

Problems Involving Rotation

Clearly specify the rotation axis.
Specify the direction of positive rotations about the axis, particularly if you are taking clockwise to be positive.

Graders always appreciate clarity, so specifically indicating that up is positive in a problem involving gravity or that counterclockwise is positive in a problem involving rotation is never considered "overkill".