The physics
The junction law. Charge does not pile up at a point. The currents into a point add up to the currents out of it.
ΣIin = ΣIout
The loop law. Walk once round a closed loop and add every change in potential. You come back to where you started, so the total is zero.
ΣE = ΣIR
A cell crossed from − to + is a rise of E. A resistor crossed along the current is a drop of IR. Against the current, it is a rise.
Several branches may join the same two points. Call one point 0 V and the other V. Then the junction law gives V.