What is the definition of the well-founded part of a model of set theory? I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the well-founded part of a model? If someone could give me a precise definition (maybe it can be defined using transitive closures, but I don't really know) of the well-founded part of a model, it'd be greatly appreciated.
Addendum
The well-foundedness that I'm referring to is not the internal well-foundedness that comes from assuming the Axiom of Regularity within the model. It's an external property, as viewed from outside the model.
 A: Suppose that $(M,E)$ is a model of ZFC, this is a set in the universe (which is also a model of ZFC, for our purposes).
It is possible that $(M,E)$ is not a well-founded relation. Internally, of course, this is impossible. $M$ does not have any element which is a decreasing sequence in $E$, since $M$ satisfies the axiom of regularity.
However we, as educated men staring at $M$ externally, know that it is possible that $M$ has more than it knows about. One can now ask about the ordinals of $M$. Namely $(Ord^M,E)$ as a linear order. This order has a maximal initial segment which is well-founded.
The well-founded part is the initial part [internally] of $(M,E)$ which is truly well-founded. It is exactly the sets whose [internal] von Neumann rank is an ordinal in the well-founded part of $(Ord^M,E)$.
A: This is a definition taken from the proof of Theorem 47 of Azriel Lévy’s monograph, A Hierarchy of Formulas in Set Theory (Memoirs of the AMS, Number 57).

Definition. Let $ M $ be a set, and $ E $ a binary relation on $ M $.
  
  
*
  
*A subset $ X $ of $ M $ is called $ E $-transitive if and only if
  $$
(\forall x,y \in M)(((y \in X) \land ((x,y) \in E)) \to (x \in X)).
$$
  
*The $ E $-transitive closure of an element $ x $ of $ M $ is defined as the
  following subset of $ M $:
  $$
\{
y \in M \mid
(\forall X)
(
((x \in X \subseteq M) \land (X ~ \text{is} ~ E \text{-transitive})) \to
(y \in X)
)
\}.
$$
  
*A subset $ X $ of $ M $ is called $ E $-well-founded if and only if for
  every non-empty subset $ Y $ of $ X $, there exists a $ y \in Y $ such that
  $ (x,y) \notin E $ for every $ x \in Y \setminus \{ y \} $.
  
*The $ E $-well-founded part of $ M $ is finally defined as the following
  subset of $ M $:
  $$
\{
x \in M \mid
\text{The} ~ E \text{-transitive closure of} ~ x ~ \text{is} ~ E
\text{-well-founded}
\}.
$$
  

