coalgebra
An ordinary algebra lets you take two things and combine them into one — multiplication merges. A coalgebra does the mirror image: it lets you take one thing and split it into a sum of pairs. If you imagine running the arrows of an algebra backwards, you get a coalgebra. This 'reverse the arrows' philosophy is what the prefix 'co' always signals, and it turns out to be enormously productive.
Formally, over a field k, a coalgebra is a vector space C equipped with two linear maps: a comultiplication Δ : C -> C ⊗ C and a counit ε : C -> k. These must satisfy axioms dual to associativity and unitality. Coassociativity says (Δ ⊗ id) ∘ Δ = (id ⊗ Δ) ∘ Δ as maps C -> C ⊗ C ⊗ C, and the counit axiom says (ε ⊗ id) ∘ Δ = id = (id ⊗ ε) ∘ Δ after the canonical identifications k ⊗ C ≅ C ≅ C ⊗ k.
The duality is precise: if A is a finite-dimensional algebra, then its linear dual A* = Hom(A, k) is naturally a coalgebra, because the dual of the multiplication A ⊗ A -> A is a map A* -> (A ⊗ A)* ≅ A* ⊗ A*. In infinite dimensions one must be careful, since (A ⊗ A)* is generally larger than A* ⊗ A*, and the dual of an algebra is only a coalgebra after passing to a suitable restricted dual.
Let C have basis {g} with Δ(g) = g ⊗ g and ε(g) = 1. Coassociativity reads (g ⊗ g) ⊗ g = g ⊗ (g ⊗ g), which holds, and the counit axiom reads ε(g) g = g. This one-dimensional coalgebra is the linear dual of the group algebra of the trivial group.
The smallest nontrivial coalgebra: a single grouplike element.
A coalgebra need not be cocommutative (the analogue of commutative), just as algebras need not be commutative. The Fundamental Theorem of Coalgebras says every coalgebra is the directed union of its finite-dimensional subcoalgebras — coalgebras are 'locally finite,' which has no algebra analogue and reflects that Δ never enlarges support.