Mathematicians are champions at taking words and giving them technical definitions, some of which make sense compared to ordinary usage, and many of which do not. From the Latin modus, meaning a “measure, manner, style, or way” we get the words:
modulus: the “base” (or “measure”) of a congruence in number theory
modulus: the absolute value of a complex number (its “measure”)
module: like a vector space, but with rings (another adaptation) instead of fields (ditto).
modular lattice: a lattice which satisfies (x∧y)∨(x∧z)=x∧(y∨(x∧z))
modular function and modular form: a complex-valued function satisfying complicated conditions hard to summarize, see Modular Function and Modular Form.
mode: the most frequently appearing element of a sample, another "measure"
A StackExchange answer tracing some of the usages: What is the origin of the term "modular" in different areas? One connection listed there is that in 19th-century German mathematics “modul” was used for what we now refer to in English as a “lattice”.
The use of the word “module” and “modular” to mean design with interchangeable or composable units is newer than these mathematical usages. “Modular” in this sense was first recorded 1936, and “module” as a component was first recorded 1955. This usage may not come from mathematics at all, but from an architectural definition module | building, a unit picked to make consistent dimensions for parts of the building.
Originally answered on Quora: https://www.quora.com/How-do-we-relate-modular-design-to-mathematics/answer/Mark-Gritter