I often think of "universal examples". This is useful because then you can actually prove something in the general case - at least theoretically - just by looking at these examples.

Semigroup: $\mathbb{N}$ with $+$ or $*$

Group: Automorphism groups of sets ($Sym(n)$) or of polyhedra (e.g. $D(n)$).

Virtual cyclic group: Semidirect products $\mathbb{Z} \rtimes \mathbb{Z}/n$.

Abelian group: $\mathbb{Z}^n$

Non-finitely generated group: $\mathbb{Q}$

Divisible group: $\mathbb{Q}/\mathbb{Z}$

Ring: $\mathbb{Z}[x_1,...,x_n]$

Graded ring: Singular cohomology of a space.

Ring without unit: $2\mathbb{Z}$, $C_0(\mathbb{N})$

Non-commutative ring: Endomorphisms of abelian groups, such as $M_n(\mathbb{Z})$.

Non-noetherian ring: $\mathbb{Z}[x_1,x_2,...]$.

Ring with zero divisors: $\mathbb{Z}[x]/x^2$

Principal ideal domain which is not euclidean: $\mathbb{Z}[(1+\sqrt{-19})/2]$

Finite ring: $\mathbb{F}_2^n$.

Local ring: Fields, and the $p$-adics $\mathbb{Z}_p$

Non-smooth $k$-algebra: $k[x,y]/(x^2-y^3)$

Field: $\mathbb{Q}, \mathbb{F}_p$

Field extension: $\mathbb{Q}(i) / \mathbb{Q}, k(t)/k$

Module: sections of a vector bundle. Free <=> trivial. Point <=> vector space.

Flat / non-flat module: $\mathbb{Q}$ and $\mathbb{Z}/2$ over $\mathbb{Z}$

Locally free, but not free module: $(2,1+\sqrt{-5})$ over $\mathbb{Z}[\sqrt{-5}]$

... perhaps I should stop here, this is an infinite list.