Node voltage analysis is one the most commonly used methods for analyzing circuits. With this method, given the value of a circuit's resistors and voltage/current sources, you can find the voltage at each node and the currents through each branch.
A node is where a circuit path splits into two or more paths, like a copper wire being split in two. The copper wire in the picture has a current $i_1$ entering a node from which two wires with currents $i_2$ and $i_3$ emerge.
Kirchoff's Current Law (KCL) says that the current entering a node, is equal to the sum of currents leaving the node. So in the copper wire, $i_1=i_2+i_3$
The next picture shows two equivalent circuits to illustrate the way to think of nodes in a rectangular circuit.
The node voltage analysis method starts by selecting a reference node, usually ground, which we'll set to 0. Then you label the current in each branch and the voltage at each node. Using Ohm's law to write an equation for every current and voltage, we end up with a system of equations we can use to solve for all unknowns. For example:
$i_1=10mA$; $i_2=50mA$; $R_1=1k$; $R_2=2k$; $R_3=10k$; $R_4=2k$