transfinite arithmetic, cardinal arithmetic, ordinal arithmetic
prime field, p-adic integer, p-adic rational number, p-adic complex number
arithmetic geometry, function field analogy
A sum is a result of an operation called addition and denoted $+$ (binary), $0$ (nullary), or $\sum$ (arbitrary).
A group (or similar algebraic structure) $A$ is written additively if its operation is a sum: $a,b \in A \;\vdash\; (a + b) \in A$. Examples include addition of natural numbers (an abelian monoid) and the generalisation to other kinds of numbers (most of which form abelian groups or at least abelian monoids, although the ordinal numbers form a nonabelian monoid).
In the case of numbers or more generally any topological abelian group or topological vector space (and generalizations), we can consider sums of infinite series, and more generally integrals. (There are however also noncommutative integrals when the order of summation/multiplication of noncommuting quantities is taken into account.)
In a category the term sum may refer to the coproduct of two objects; in particular, the sum of two abstract sets is their disjoint union. The sum of objects in a preadditive category may refer to the biproduct or direct sum.
In a poset this becomes the notion of join; in particular, the union of two concrete sets (material sets or subsets of a fixed set $U$) was once often called their sum.
A more general notion of sum in category theory and type theory is that of the dependent sum of a family of objects.