Defintion of equivalent circuit
Equivalence of circuits means that if each was hidden in a black box (no other information from inside) with only their terminals exposed you could not discriminate between them; no matter what you connect to the terminals.

If you provide a certain voltage to the terminals the current through the terminals  will be the same and and vice versa, i.e. if you let flow a certain current through the terminals the voltage across the terminals will be the same for both circuits if they are equivalent.

Note: It does not mean, however, that they would dissipate the same amount of power, i.e. dissipated heat must also be thought to be hidden inside the black box (e.g. a Norton source and a Thevenin source may be equivalent, but in open circuit configuration only the Norton source would create heat; the Thevenin source not).

If it is known that two circuits are linear circuits (with 2 terminals) there are some simple criteria to determine whether they are equivalent or not:  


E.g.: If their open circuit voltage and their short circuit current are both equal then the two linear circuits are equivalent.
Or more general: 
if their V-I-characteristics (which are V-I-lines) have two points in common then they are equivalent.