The relation $R=\{(1,1),(2,2),(3,3)\}$ on the set $\{1,2,3\}$ is

Let $A = \{1, 2, 3\}$ and $R = \{(1, 1), (2, 2), (3, 3)\}$.

A relation $R$ is reflexive if $(a, a) \in R$ for all $a \in A$. Since $(1, 1), (2, 2), (3, 3) \in R$, the relation is reflexive.

A relation $R$ is symmetric if $(a, b) \in R \implies (b, a) \in R$. For all elements in $R$, $(a, b) = (b, a)$, so the condition holds trivially. Thus, the relation is symmetric.

A relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$. Since the only pairs are of the form $(a, a)$, if $(a, a) \in R$ and $(a, a) \in R$, then $(a, a) \in R$, which is true. Thus, the relation is transitive.

Since $R$ is reflexive, symmetric, and transitive, it is an equivalence relation.

Answer: an equivalence relation