For students who are in multivariable (MV) calculus, one of the main topics you study has to do with the vector differential operator called del (the upside down triangle). Using this operator we can define gradient, divergence, and curl. Another useful operator for waves, E&M, quantum mechanics, and other physics topics, is the laplacian (del-squared).
In physics, gradients can be used to describe vectors that are directed from high values of a scalar quantity to low values. An example is the force of gravity, which is directed to move objects from high potential energy to low potential energy. We have electric fields flow from high electric potential (voltage) towards low electric potential. We see heat flow from high temperature towards low temperature, or air masses flow from high pressure towards low pressure. These can all be described by gradients.
Divergence is used for vector fields that radiate from a point source. This could be light, sound, radiation, gravitational fields, or electric fields, all radiating from point sources.
Curl is used for vector fields that circulate around some vector source. A prime example is the circulating magnetic fields that are found around electric currents, or circulating electric fields around a changing magnetic field.
Check out the video to see a few examples, and hopefully you will see how these strange mathematical operators are applied to real, physical phenomena.