Are non-standard models always not well-founded? Are non-standard models of ZF set theory by definition always not well-founded? And it seems it is, because it must be. 
But then, Wikipedia says that when there is a set that is a standard model of ZF, there exists smallest set $L_k$ that is standard model. So, this seems to confuse me a lot.

By a standard model of ZF we mean a model $( M , E )$ of ZF where the relation $E$ is real membership on $M$, i.e., $x \mathrel{E} y \; \Leftrightarrow \; x \in y$.  The relation $E$ is usually denoted either simply by $\in$, or by $\in_M$ to emphasise that we've restricted the domain.
Given any standard model $( M , \in_M )$ of ZF we can construct an isomorphic non-standard model $( M^* , E^* )$ quite easily by taking $M^* = \{ \{ x \} : x \in M \}$, and defining $\{ x \} \mathrel{E^*} \{ y \} \; \Leftrightarrow \; x \in y$.  This model will be well-founded.
