Sometimes, coming up with good mathematical notation is key to understanding parts of mathematics. For example, consider the quadratic formula.
Brahmagupta formulated a version of the quadratic formula in 628 A.D., which goes as follows:
To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value.
Compare this to our modern notation:
$$ \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$
Not only is the modern notation more succinct, it also makes certain properties of quadratic equations more apparent. For example, it's easy to determine the number of real roots of a quadratic equation using the modern formula. But Brahmagupta's formula is more opaque, and it's not immediately obvious how to find the number of real roots.
A second example is the introduction of the digit 0 and the place value system. When doing complicated arithmetic, Roman numerals are difficult to manage. But number systems with place value systems can handle addition much more intuitively.
What I am looking for is specific examples of mathematical notation, where its introduction accelerated progress in that particular field. I'm especially interested in more elementary mathematics--i.e. everything before the 1700s--but examples in more modern math are welcome too.