The intention of this question is to find practical examples of improved mathematical notation that enabled actual progress in someone's research work.
I am aware that there is a related post Suggestions for good notation. The difference is that I would be interested especially in the practical impact of the improved notation, i.e. examples that have actually created a better understanding of a given topic, or have advanced actual research work on a given topic, or communication about results.
I would be interested in three aspects in particular
(1) Clarity and Insights: Improved and simplified notation that made structures and properties more clearly visible, and enabled insights for the researcher.
(2) Efficiency and Focus: Notation that created efficiencies (e.g., using less space and needed less time, dropped unnecessary or redundant details).
(3) Communication and Exposition: Improved notation that supported communicating and sharing new definitions and results. And notation that evolved and improved in the process of communication. Would you have any practical examples of this evolving process, including dead-ends and breakthroughs?
Edit: Have received great examples in the answers that illustrate what I am interested in. Very grateful for that!
Best Answer
There is a notation that had an immediate and profound impact on research in algebraic topology, later algebraic geometry, and was eventually adopted by all areas of mathematics: the introduction of arrows to denote mappings. Compare $f \colon X \to Y$ with $f(X) \subset Y$, which is what was used previously. It meets all three criteria mentioned by the OP and is recognized by every mathematician.
Just as importantly, the use of arrows led to commutative diagrams, without which many parts of modern mathematics are now inconceivable. I mentioned this before in an answer here.