According to Wikipedia, Carl Friedrich Gauss invented it. Quote Wikipedia,

"Modular arithmetic was introduced by Carl Friedrich Gauss in his book Disquisitiones Arithmeticae, published in 1801."

In ordinary arithmetic, the resulting value will be from an infinite set of values but in case modular arithmetic, resulting value will be from a finite set of values.

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e.g. in ordinary arithmetic, the sum of two integers, the result will be an integer in the range {..., -3, -2, -1, 0, 1, 2, 3, ...}

for modular arithmetic, the sum of two integers

like (a+b)mod n will be in the range {0, 1, 2, ... n-1}