In $\mathbf{Set}$, the product $A \times B$ is the Cartesian product $\{(a,b) : a \in A, b \in B\}$ with the usual projections. More generally, $\prod_{i \in I} A_i = \{(a_i)_{i \in I} : a_i \in A_i\}$ is the set of all choice functions.

In $\mathbf{Set}$, the coproduct $A \sqcup B$ is the disjoint union $A \sqcup B = \{(a, 0) : a \in A\} \cup \{(b, 1) : b \in B\}$. Given $f : A \to X$ and $g : B \to X$, the unique map $[f,g]$ acts as $f$ on elements from $A$ and $g$ on elements from $B$.

In $\mathbf{Grp}$, the product $G \times H$ is the direct product with componentwise operations: $(g_1, h_1) \cdot (g_2, h_2) = (g_1 g_2, h_1 h_2)$. The projections are the standard group homomorphisms.

In $\mathbf{Grp}$, the coproduct is the free product $G * H$: words alternating between elements of $G$ and $H$, subject only to the relations in $G$ and $H$ individually. For example, $\mathbb{Z} * \mathbb{Z} = F_2$ (the free group on 2 generators). The free product is typically much larger than the direct product.

In $\mathbf{Ab}$, finite products and finite coproducts coincide: $A \times B \cong A \oplus B$ (the direct sum). The projections $A \oplus B \to A$ and inclusions $A \hookrightarrow A \oplus B$ make it both a product and a coproduct. For infinite families, the product $\prod_i A_i$ and coproduct $\bigoplus_i A_i$ differ: the coproduct consists of tuples with only finitely many nonzero entries.

In $\mathbf{Top}$, the product $X \times Y$ has the product topology (generated by products $U \times V$ of open sets). For arbitrary products $\prod_i X_i$, this is the product topology (coarsest topology making all projections continuous), not the box topology.

In $\mathbf{Top}$, the coproduct $\bigsqcup_i X_i$ is the disjoint union with the topology where $U \subseteq \bigsqcup_i X_i$ is open iff $U \cap X_i$ is open in $X_i$ for each $i$.

In a poset $(P, \leq)$ viewed as a category, the product of $a$ and $b$ is the greatest lower bound (meet, infimum) $a \wedge b$. The coproduct is the least upper bound (join, supremum) $a \vee b$. A category where all binary products and coproducts exist corresponds to a lattice.

In $\mathbf{Sch}$, the product $X \times_S Y$ (fiber product over the base $S$) is the scheme representing $T \mapsto \mathrm{Hom}(T, X) \times_{\mathrm{Hom}(T, S)} \mathrm{Hom}(T, Y)$. For affine schemes, $\mathrm{Spec}\, A \times_{\mathrm{Spec}\, R} \mathrm{Spec}\, B = \mathrm{Spec}(A \otimes_R B)$.

In $R\text{-}\mathbf{Mod}$, the product $\prod_{i \in I} M_i$ is the direct product (all tuples) and the coproduct $\bigoplus_{i \in I} M_i$ is the direct sum (finite support). When $I$ is finite, these coincide.

In $\mathbf{CRing}$, the coproduct is the tensor product: $A \sqcup B = A \otimes_{\mathbb{Z}} B$. This is because $\mathrm{Hom}(A \otimes_{\mathbb{Z}} B, C) \cong \mathrm{Hom}(A, C) \times \mathrm{Hom}(B, C)$ naturally. In $\mathbf{CRing}/R$ (the category of $R$-algebras), the coproduct is $A \otimes_R B$.

Not every category has products. In the category of fields $\mathbf{Field}$, the product $\mathbb{Q} \times \mathbb{F}_p$ does not exist. If it did, there would be ring homomorphisms from a field to both $\mathbb{Q}$ and $\mathbb{F}_p$, which is impossible since they have different characteristics.