The modulo operator, as in a mod b, was introduced by Carl Friedrich Gauss in Disquisitiones Arithmeticae:

• https://la.wikisource.org/wiki/Disquisitiones_arithmeticae

written in 1798, and published in 1801. Writing in Latin, he naturally used a Latin term, "modulo", meaning "by a measure of". Thus, a mod b can be read as a by a measure of b.

This wasn't quite the origin of modular arithmetic, since Euler used modular arithmetic to simplify the solution of remainder problems in 1734 (Bullynck 2009). But it was not until Gauss that the modulo operator was introduced, and Gauss gave the first thorough treatment of modular arithmetic.

Remainder problems, which are closely related, are much older, with old roots in ancient Chinese mathematics (the Chinese Remainder Theorem) and ancient Greek mathematics (Euclid and Diophantus), and early medieval Indian mathematics (Bullynck 2009).

The similarly-named modulus function (i.e., the absolute value function) has a similar etymology, with the original term being the French "module" meaning "measure". This was introduced in 1806 by Jean-Robert Argand, in the context of the absolute value of complex numbers. The concept of absolute value is much older, but if one is only considering negative and positive number, very simple. "Module" imported into English was Latinised to "modulus". Our modern notation and "absolute value" came later, the notation introduced by Weierstrass in 1841.

Reference:

Maarten Bullynck, "Modular arithmetic before C.F. Gauss: Systematizations and discussions on remainder problems in 18th-century Germany", Historia Mathematica 36(1), 48-72 (2009)