Regular. To start off:
The regular representation of a group $G$ over a field $k$ is the action on $k[G]$ given by group multiplication.
A topology is regular if a closed set and a point not in that set can be separated by disjoint open sets.
A point $\zeta_0$ on the boundary of a domain in $\mathbb C$ is called regular if there exists a subharmonic barrier function $b(z)$ defined within $D$ near $\zeta$. This may not be the standard definition but Gamelin's complex Analysis defines it as a subharmonic function $\omega(z)$ on ${|z-\zeta_0|<\delta}\cap D$ which is negative everywhere, tends to 0 at $\zeta_0$, but $\limsup(\omega(z))<0$ as $z$ tends to any other boundary point of $D$ within distance $\delta$ of $\zeta_0$.
I've borrowed/paraphrased the following from the Wikipedia disambiguation page but removed a couple that either are not too relevant to pure math or qualify the "regularity" more. Feel free to put them in too.
Regular cardinal, a cardinal number that is equal to its cofinality
Regular element, and regular sequence and regular immersion.
Regular code, an algebraic code with a uniform distribution of distances between codewords
Regular graph, a graph such that all the degrees of the vertices are equal
The regularity lemma, which has nothing to do with regular graphs
Regular polygon, and regular polyhedron
Regular prime: a prime $p$ that does not divide the class number of the $p$th cyclotomic field $\mathbb Q[\zeta_p]$.
Regular surface in algebraic geometry
Regularity of an elliptic operator
JS Milne's comment: A regular map is a morphism of algebraic varieties.
Regular value of a differentiable map
Regular ring (Note: this definition can be made noncommutative. A right noetherian ring R is said to be right regular if every finitely generated right R-module has finite global dimension. See Lam's Lectures in Modules and Rings, Section 5G.)
Regular language, a language that can be accepted by a finite state machine.
Absolutely regular is a synonym for $\beta$-mixing in stochastic processes.
Regular matroid, a matroid which is representable over every field. In this sense, all graphs are regular (their cycle matroids are regular), which has nothing to do with regular graphs.
