The junction rule says that the sum of the currents going into a junction must equal the sum of the currents leaving a junction. This describes the conservation of current.
"Kirchhoff's Junction Rule" and "Kirchhoff's Current Law" are the same thing.
The rule is that the amount of current flowing IN to any point in a circuit is equal to the amount of current flowing OUT of the same point. EVEN if the point is a complicated 'junction' with a thousand different paths in and out of it !
This whole idea is pretty obvious if you understand the law of Conservation of Electric Charge. Charge can't be created out of nothing, and it can't disappear or be destroyed. So whenever you see charge appear, it had to come from somewhere; and whenever you see charge disappear, it had to go somewhere.
That's exactly what's happening at the 'junction' in an electric circuit. Current into the junction brings charge into it. The charge can't pile up there, and it can't just disappear, so it has to leave the junction in a current that's exactly the same size as the current that brought it there.
The "junction rule" (Kirchhoff's first law) says that the value of the electric current entering any point in the circuit must be the same as the one leaving it. This comes as a consequence of the conservation of charge, the electric charge that comes in must come out (when speaking about currents, per unit time).