Can be viewed as a baby version of derived equivalence, when we say two algebras \(A\) and \(B\) are Morita equivalence, they are equivalent in a deep sense. Precisely speaking it means their module categories are equivalent.
\[ {}_{}\mathbf{Mod}_A \cong {}_{}\mathbf{Mod}_B \]
When you have a full idempotent \(e\) in \(A\), i.e. an idempotent \(e^2=e\) that satisfy \(AeA=A\), here the left side accepts linear extending. Then \(A\) is automatically Morita equivalent to \(B=eAe\), the functor is given by
\[ E : {}_{}\mathbf{Mod}_A \to {}_{}\mathbf{Mod}_B, \quad M\mapsto Me \]
with reverse functor given by
\[ F : {}_{}\mathbf{Mod}_B \to {}_{}\mathbf{Mod}_A, \quad N\mapsto N\otimes_B A.\]
Note that here \(Me\) is a right \(B=eAe\) module because \(meae=(mea)e\).