What is the name of the law that states that the sum of currents entering a junction equals the sum leaving?

The law stating that the sum of currents entering a junction equals the sum leaving is Kirchhoff's current law.

It is also called Kirchhoff's first law or the junction rule. At any node where circuit branches meet, current is conserved: the algebraic sum of all currents is zero. For example, if 5 amperes enter a junction and 2 amperes leave through one branch, the remaining branch must carry 3 amperes away, assuming no charge accumulates at the node.

The law follows from conservation of electric charge. Current measures the rate at which charge flows, so a junction cannot continuously create or destroy charge under the ordinary lumped-circuit assumptions. Engineers often write it as ΣI = 0, assigning positive signs to currents entering and negative signs to currents leaving, or vice versa.

Kirchhoff's voltage law is different: it says the directed sum of voltage changes around a closed loop is zero. Together with Ohm's law, Kirchhoff's current law forms the basis of nodal analysis and is used by circuit simulators such as SPICE. At very high frequencies, distributed electromagnetic effects may require more advanced models.