One of the most classical examples is the word «algebra», which denotes not only a branch of mathematics, but also the following mathematical objects:

In linear algebra an algebra is a vector space, equipped with a bilinear operator (called product).

In set theory an algebra is a collection of sets closed under finite union, finite intersection and complement.

In universal algebra an algebra is a set, equipped with collection of finitary operations.

Note, that any algebra in linear algebraic sense is also an algebra in universal algebraic set, but not vice versa.

A quite similar thing happens with varieties:

In algebraic geometry a variety is the set of solutions of a system of algebraic equations.

In universal algebra a variety is a class of all algebras (in universal algebraic sense) with a given signature, satisfying a given set of identities.

Moreover, both those «varieties» are translated to Russian as «многообразия» - the same word, that is used for manifolds (topological spaces, such that each point of them has a neighbourhood, that is homeomorphic to $\mathbb{R}^n$ for some fixed $n$)

And if I have reached the theme of ambiguous translations, I think, that I should mention that «perfect groups» (groups equal to their derived subgroup), «complete groups» (centerless groups isomorphic to their automorphism group) and «immaculate groups» (finite groups, whose order is equal to the sum of orders of their proper normal subgroups) are all translated to Russian as «совершенные группы».

Also, the following examples deserve to be mentioned:

Two abstract groups are called commensurate (or commensurable), if they have isomorphic subgroups of finite index. Two subgroups of a group are called commensurate (or commensurable) if their intersection has finite index in both of them. Note, that two subgroups of a group may be commensurate as abstract groups, but not commensurate as subgroups.

Artinian groups are groups, that satisfy the minimum condition on subgroups. Artin groups are groups, that have a presentation of specific form. Both of them are «Артиновы группы» in Russian.

A right (left) ideal of a ring is a subring, that is closed under right (left) multiplication on arbitrary element of the ring. A right (left) ideal of a semigroup is a subsemigroup, that is closed under right (left) multiplication on arbitrary element of the semigroup. Note, that an ideal of the multiplicative semigroup of an associative ring is not always an ideal of that ring (because it does not need to be closed under addition).

Cubic graphs are usually defined as finite simple 3-regular graphs. However, the Hamming graph $H(3, 2)$ is also referred by some authors as «The Cubic Graph». Well, it is indeed finite, simple and 3-regular, but not the only one with this property.

The definition of simple graph I am used to is "graph without loops and multiple edges", however I know, that some people define simple graph as "graph without multiple edges" (loops are allowed).

In different sources $D_{2n}$ means either $C_{2n} \rtimes C_2$ or $C_n \rtimes C_2$.

The word "automaton" is sometimes used as a synonym for "acceptor" and sometimes for "transducer"

Sometimes $C_n^k$ means binomial coefficient $\frac{n}{(n - k)!k!}$ and sometimes direct product of $k$ isomorphic copies of a cyclic group of order $n$.