Kirchhoff's laws and circuit analysis

Kirchhoff's two laws are conservation statements dressed as circuit rules. The current law says that whatever flows into a junction flows out of it, because charge cannot pile up there. The voltage law says that the potential changes around any closed loop sum to zero, because coming back to where you started must return you to the same potential. Between them they solve any network of sources and resistors, however tangled.

The current law kills the most persistent beginner's picture, in which current is a fuel consumed by the components it passes through. It is not. Exactly as many electrons leave a bulb as enter it; what the bulb consumes is energy, and that shows up as a drop in voltage, not a drop in current. A resistor in a series loop has the same current on both sides and a different potential on each.

The voltage law dismantles the other one — "current takes the path of least resistance". Current takes every available path, in inverse proportion to each path's resistance. Both parallel branches see the same potential difference, so both carry current; the lower-resistance branch simply carries more. The phrase is only true for the extreme case of a short circuit, where one path's resistance is so much lower that everything else is negligible by comparison, and that special case has done a lot of damage as a general rule.