In the above two images, the scalar field is in black and white, black representing higher values, and its corresponding gradient is represented by blue arrows.
The English Wiktionary has a dictionary definition (meanings of a word) for: gradient

In vector calculus, the gradient of a scalar field is a vector field which points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.

A generalization of the gradient, for functions which have vectorial values, is the Jacobian.

Consider a room in which the temperature is given by a scalar field $\phi$, so at each point $(x,y,z)$ the temperature is $\phi(x,y,z)$. We will assume that the temperature does not change in time. Then, at each point in the room, the gradient at that point will show the direction in which the temperature rises most quickly. The magnitude of the gradient will determine how fast the temperature rises in that direction.
Consider a hill whose height above sea level at a point $(x, y)$ is $H(x, y)$. The gradient of $H$ at a point is a vector pointing in the direction of the steepest slope or grade at that point. The steepness of the slope at that point is given by the magnitude of the gradient vector.