Recently, I realized that there are some famous problems in mathematics whose solution depended heavily on the right formulation of an intuitive concept.
For example, there was no precise definition of an algebraic integer before Dedekind. As Milne says in his book on algebraic number theory, Euler's proof of Fermat's Last theorem for the exponent $3$ does only become correct when you replace $\Bbb Z[\sqrt{-3}]$ by $\Bbb Z[\frac{1+\sqrt{-3}}{2}]$.
Do you know any other examples of theories where the correct formulation of a concept was an important step in the evolution of that theory?
(As usual, one example per answer.)