Mesh analysis is a technique that can be used to solve for the voltages and currents in a planar circuit. A planar circuit is one that may be drawn with no wires overlapping another. Mesh analysis is a systematic algorithm that may always be used on planar circuits. A circuit with N loops will yeild a system of N equations to solve simultaneously. Mesh analysis is based on Kirchhoff's current law and voltage law.
The current law states that all currents entering a node sum to zero (i.e. conservation of charge). The voltage law states that for any closed loop, the voltage drops across the components (resistors, etc.) of that loop sum to zero (i.e. conservation of energy).
For three loops, the first step is to label each loop, i.e. 1, 2, 3, and define a mesh current for each loop. Each current should rotate in clockwise in the loop. For each electrical component in the loop (going in order to keep the polarity consistent), determine the voltage drop due to that mesh current. For each component other than current or voltage sources, the voltage drop is the impedance times the mesh current. Each of the voltage drops in the loop should sum to zero.
Current sources provide an exception in that they fix the mesh current at the value of the current source. So for loops with current sources, set the loop current equal to the current source value. If a current source spans two or more loops, then the mesh currents should add (or subtract) to equal the current source.