Meaning of iterable model In "Strong axioms of infinity and the search for V", Woodin states that Projective Determinacy holds if and only if for all finite ordinals $n$ there is a countable iterable model for ZFC plus $n$ Woodin cardinals. 
What is the definition of an iterable model for $n$ Woodin cardinals? 
 A: A model is iterable if repeatedly taking ultrapowers never results in an ill-founded model.
There are several wrinkles formalizing this. For example, ultrapowers are with respect to extenders rather than simply ultrafilters, and the ways in which direct limits are to be computed can get quite involved. The reason for the former is because we want the embeddings to witness significant large cardinal assumptions (versions of strongness, for instance). 
The reason for the latter is more complicated, but at the end of the day traces back to the desire for a successful comparison process. For models with decent large cardinal structure (in particular, for models with Woodin cardinals), linear comparisons as for the $L [\mu] $ models do not suffice. Attempting to solve this difficulty led to the identification of the concept of iteration tree. The modern formulation of iterability, based on this concept, involves a certain game, where ultrapowers are repeatedly taken for various models so that at limit stages several possible direct limits can be considered, and the model is iterable if and only if there is always a way of picking a well-founded limit so the process can continue.
In slightly more detail, you form a tree structure whose nodes are models, with the model you begin with as the root, and with the property that if a model $M'$ is placed as a successor of a model $M$, then it results from taking an ultrapower of $M $. The tree is built by stages. At each nonlimit stage, an extender from the latest model added to the tree is chosen, and a model is picked among the nodes of the tree to form its ultrapower via this extender. This ultrapower is then added to the tree and the stage ends. At limit stages, a certain branch of the tree so far is selected, and the direct  limit of the models along the branch is chosen as the new model to add, on top of the branch. This can be seen as a game. Player I picks extenders and models at nonlimit stages, player II picks branches at limit stages. Player II loses if the resulting direct limit is ever illfounded. The game continues for a certain number of stages $\kappa $. If player II has a winning strategy, the bottom model is called $\kappa$-iterable. There are several variants of this iteration game, resulting on various notions of iterability.
I recommend Steel's handbook chapter for the precise definitions and appropriate context.
