TITLE: What definitions were crucial to further understanding?
QUESTION [53 upvotes]: Often the most difficult part of venturing into a field as a researcher is to come up with an appropriate definition. Sometimes definitions suggest themselves very naturally, as when you solve a problem and then ask, ‘What if I generalize this a bit?’
Other times they arise only after struggling with a subject and realizing you were looking at it from the wrong angle. An appropriate definition can then make all the difference, by reorganizing the thought and sheding light into the problems, somehow making them sharper and more focused. 
I would like to collect evidences and instances of this idea. An answer should be a story of how someone came up with a good definition and how this was crucial to their understanding of a topic. If you talk about someone else, then ideally provide a reference.
(My interest in this is mainly psychological, namely, how the action of naming something somehow brings it into existence and organizes the world around it.)
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It was suggested that this question is a duplicate of (Examples of advance via good definitions), which asks about good definitions. It was pointed out there, and also here, that basicaly all notions that stood the test of time qualify as good definitions.
I was not asking, although it seems many people construed it this way, for a collection of good definitions, but for a collection of stories that showed how a proper definition actually changed the perception about a field. 
A typical such story would have someone saying "Wait a moment! I should not be dealing with [concept A] at all! That's the wrong way to approach this. Instead I should define this other guy, [concept B], and then everything will make a lot more sense!"
I realize this is hard to make precise, so I understand if the question gets closed.

REPLY [5 votes]: I want to add the notion of Tree-Width to the list.
Roughly speaking, tree-width is a measure of how less tree-like a graph is. A tree has tree-width 1 while a $k$-clique has tree-width $k$. Tree-width has been used as a parameter for the analysis of a number of graph algorithms.
I think like every other clever definition, the definition of tree-width is something that makes a lot of connections clearer once we internalize it and yet, the definition is not something that is easy to come up with.
For reference,

A tree decomposition of a graph $G = (V, E)$ is a tree, $T$, with nodes
  $X_1$, ..., $X_n$, where each $X_i$ is a subset of $V$, satisfying the following
  properties (the term node is used to refer to a vertex of $T$ to avoid
  confusion with vertices of $G$): 

The union of all sets $X_i$ equals $V$. That is, each graph vertex is contained in at least one tree node.
If $X_i$ and $X_j$ both contain a vertex $v$, then all nodes $X_k$ of $T$ in the (unique) path between $X_i$ and $X_j$ contain $v$ as well. Equivalently, the
  tree nodes containing vertex $v$ form a connected subtree of $T$.
For every edge $(v, w)$ in the graph, there is a subset $X_i$ that contains both $v$ and $w$. That is, vertices are adjacent in the graph
  only when the corresponding subtrees have a node in common.

The width of a tree decomposition is the size of its largest set $X_i$
  minus one. The treewidth $\mathrm{tw}(G)$ of a graph $G$ is the minimum width among
  all possible tree decompositions of $G$.