The differential form of Maxwell's equations describes the dependence of the electric and magnetic fields point-by-point. Equation one shows that the divergence of the electric field at a point is proportional to the free charge density at that point. Equation four shows that the curl of the magnetic field at a point is the sum of a term proportional to the conduction current density jc at that point and a term proportional to the partial derivative of the electric field with respect to time at that same point.
Note that the first two equations are scalar. The final two equations are vector, so that they each actually represent three equations, one for each component. In differential form, there are actually eight Maxwells's equations!
The del operator, defined in the last equation above, was seen earlier in the relationship between the electric field and the electrostatic potential. In that case, the del operator acting on a scalar (the electrostatic potential), yielded a vector quantity (the electric field). The "cross product" of the del operator and a vector yeilds another vector known as the curl. The "dot product" of the del operator and a vector yields a scalar known as the divergence. The del operator is widely encountered in physics and engineering; familiarity with its properties is highly recommended!