TITLE: The Origin(s) of Modular and Moduli
QUESTION [41 upvotes]: In mathematics and in physics, people use the terms "modular..." and "moduli space" very often. I was puzzled by the etymology, the origins and the similarity/equivalence/differences for these usages/concepts behind for some time.
Could I seek for expert's explanations on the contrast/comparison of the "modular..." and "moduli space" in the following contexts:

"Modular" in the modular tensor category. (Also the "pre-modular" category.)
"Modular" in the modular form.
"Modular" in the topological modular form (tmf).
"Modular" in the modular invariance (e.g. in Conformal Field Theory).
"Moduli space" in algebraic geometry
"Moduli space" in Non-supersymetric Quantum Field Theories or gauge theories 
"Moduli space" in Supersymetric gauge theories 
"Module" in the "mod" or modular arithmetic.
"Modular" in modular representation theory of (finite) groups/algebraic groups. [As André Henriques suggested]
"Module": $G$-module in the group (co)homology and topological $G$-module for the topological group. [Likely related to 9.]
"Modulus" of convergence and "Modulus" of continuity in analysis. 

Thank you in advance for solving/illuminating this rather puzzling issue.

REPLY [7 votes]: I was similarly curious when writing the introduction of my PhD thesis,
since the context was moduli spaces of Abelian differentials.
I felt the need to dig a bit and include a discussion there.
Here is an English translation of what I wrote then. The original is
here
(see page 9 of the introduction).

1. Moduli spaces
The term "moduli space" is often preferred to "parameter space" in contexts
  where one is interested in describing geometric objects up to a certain
  equivalence.
For instance, we consider the modulus of a cylinder (ratio of its height to
  its circumference) when we are interested in its shape without caring about
  its size. The moduli space of cylinders is the set of positive reals.
A more instructive example is the moduli space of tori.
1.1. Tori.
Here again, we seek to describe the shape of a torus (of real dimension 2,
  endowed with a complex structure) without taking its size into account.
A torus can be defined as a quotient $\mathbb{C}/\Lambda$ where $\Lambda$
  is a lattice in $\mathbf{C}$. Consider tori equivalent when they correspond
  to lattices obtained from each other by rotation and dilatation.
  Given a torus, we can then consider it as  a quotient $\mathbb{C}/\Lambda$
  where $\Lambda$ is a lattice in $\mathbb{C}$ with basis $(1, \tau)$. Two
  parameters $\tau$ and $\tau'$ define equivalent tori if their difference
  is an integer, or if they are opposite or inverse of each other.
  We can therefore assume that $\lvert \operatorname{Re} \tau \rvert < 1/2$,
  $\operatorname{Im} \tau > 0$, and $\lvert \tau \rvert > 1$. This draws a
  domain in the upper half-plane, located between the vertical lines at
  x-coordinate $-1/2$ and $1/2$, and outside the circle of radius $1$
  centered at the origin. Some points in this domain still need to be
  identified: vertical half-lines at x-coordinate $-1/2$ and $1/2$ by
  horizontal translation, and the two halves of the arc of circle which
  bounds the domain below by the map $z \mapsto -1/z$. The identifications
  of these parts of the boundary of the domain can be described globally:
  they are obtained by reflection about its (vertical) symmetry axis.
We call moduli space of tori the space obtained after performing these
  identifications. Its topology is that of a sphere minus a point.
  Its geometry is richer, and is inherited from the hyperbolic metric
  on the upper half-plane. The points corresponding to $i$ and to
  $i 2 \pi / 3$ represent the square torus and the hexagonal torus,
  which respectively admit automorphisms of order 4 and 6. They are
  cone points of respective angles $\pi$ and $2\pi/3$ in the moduli
  space of tori. 
This very classical moduli space is called the modular surface or
  the modular curve, depending on whether one chooses to adopt the real
  or complex viewpoint. Because of these cone points, it is not quite
  a manifold; we call it an orbifold.

I stopped the discussion there but wish I had expanded a bit on
how Riemann found that it takes "more moduli" to describe the
geometry of closed compact Riemann surfaces of higher genus, and
concluded it took "$3g-3$ complex moduli" for genus $g > 1$.
Or if one prefers, the "modulus" of a genus $g$ closed compact
Riemann surface, for $g > 2$, is no longer a real number or a
complex number or a point in a quotient of the upper half-plane,
but a point in a $(3g-3)$-dimensional complex orbifold...