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.