Plotting

  ?PlotVectorField

  PlotVectorField[{x y,x-y},{x,-3,3},{y,-3,3}];

Think of fluid flow. The vectors represent the flow of the fluid at the tail of the vector. The length of the vector represents the speed of the flow and the direction of the vector represents the direction of the flow.

  ?PlotVectorField3D

  PlotVectorField3D[{y,-x,0},{x,-1,1},{y,-1,1},{z,0,2},
  VectorHeads->True];

  PlotVectorField3D[{Cos[x],Exp[-x] Sin[y],z-y},{x,-1,1},{y,-1,1},{z,-1,1},
  VectorHeads->True];

Up to Vector Fields