Thevenin's Theorem states that for any two linear networks A and B that are connected by two conductors but not magnetically coupled, network A may be replaced by a simpler equivalent network for the purpose of simpler circuit analysis computations with respect to network B. 

The equivalent network is known as the Thevenin equivalent network, and it consists of a voltage source Vθ(s) in series with an impedance Zθ(s). The voltage source Vθ(s) is the transform of the voltage at the two open terminals of network A, while the impedance Zθ(s) is the transform impedance at the two terminals of A with all independent sources reduced to zero.  Related to Thevenin's Theorem is Norton's Theorem, which is just the current representation of Thevenin's Theorem.

One must know the characteristics of networks to which Thevenin's Theorem is applicable.  Let A and B be the two connected but non-magnetically coupled linear networks, with current i(t) or I(s) flowing from A to B, as shown in Figure 1.  Network A must have the following characteristics:  1) it only has linear passive elements; 2) it may contain independent voltage and current sources as well as dependent (or controlled) sources; 3) it may have initial conditions present (e.g., voltages in capacitors or currents in inductors); 4) it has no magnetic coupling to B.  On the other hand, network B may be characterized as follows: 1) it has linear passive elements only; 2) it has no sources;  3) it has no initial conditions; and 4) it has no magnetic coupling to B.

Figure 1.  Thevenin Equivalent Circuit

Given the networks A and B characterized above, Thevenin's Theorem is applied by finding a network C that's equivalent to A, wherein C consists only of a voltage source Vθ(s) in series with an impedance Zθ(s), and replacing A with C. 

To find the Thevenin equivalent circuit for network A, the following steps need to be done: 1) zero out all independent sources in network A by opening all current sources and shorting all voltage sources (dependent sources are not changed); 2) determine Zθ(s) by calculating the equivalent passive impedance of network A as seen from the open terminals of A (bear in mind that all the independent sources are zeroed out at this point); and 3) apply a voltage source Vθ(s) in series with Zθ(s), with Vθ(s) chosen such that the current from C to B (once they're connected) will still be I(s). In step 2, it is assumed that A has no dependent sources, since if A has dependent sources, then Zθ(s) must be computed based on the fact that A is an active network.

Thevenin's theorem is useful in simplifying the analysis when only a part of a circuit is of interest, such as the load to an electronic circuit. As an example, Thevenin's theorem can be used to simplify the analysis on how to maximize power delivery to the load (B) of a network (A).