All of these terms are frequently used in mathematics, physics and particularly in the classical electrodynamics. These are called vector operators and they have very much physical meaning. Only problem is that it becomes hard to visualize. Let us try to understand the Physics of these operators.
First of all remember that gradient operator (grad) always operates on a scalar field, why? We answer shortly; secondly divergence (div) and curl operate only on vector field. [Please refer to this http://scienceray.com/physics/vector-potential-and-scalar-potential-made-clear/].
Consider grad operator first. Its meaning is steepness. For example temperature (scalar quantity so a scalar field) is different at different points of the room. Gradient determines this difference but there is one more thing to add into this. It is “directed” towards the greatest change in the field. This involvement of direction makes it vector quantity i.e. grad operates on a scalar field (U) and result gradU is vector field. Temperature in the room is same at every point then grad will be zero. Finally gradU is always perpendicular to the plane. Think of travelling on a plane road, you do not gain any potential energy but as soon as you start climbing a hill you start gaining P.E.
Now come to divergence. It is defined as flux per volume. As described earlier it operates on a vector field but (from definition) changes it into scalar field. Suppose there is a liquid flowing through a pipe then its divergence is zero but if it is coming out then it has non-zero divergence.
Finally curl of a vector represents circulation per unit area. [Already explained in my other article]. We can sum up and say that if divergence is zero then vector field is solenoid and if curl is zero the there is no rotation and offcourse gradU=0 means constant U.