What is Kirchoff’s current law?

Kirchoff’s current law is based on the principle of conservation of charge.This requires that the algebraic sum of the charges within a system cannot change. Thus the total rate of change of charge must add up to zero. Rate of change of charge is current.

This gives us our basic Kirchoff’s current law as the algebraic sum of the currents meeting at a point is zero. 
i.e. at a node, ΣIr= 0,  where Ir are the currents in the branches meeting at the node.

This is also sometimes stated as the sum of the currents entering a node is equal to the sum of the current leaving the node. The theorem is applicable not only to a node, but to a closed system.