Addition of quantities including numbers (scalars) and vectors is a binary operation resulting in a sum. Addition of natural numbers (positive integers) is just a matter of counting, really; to add 2 and 3 just start with two and then count three more. This fact is formalized in the Peano axioms, which define the positive integers and the operations of addition, multiplication, and induction on them as abstractions of the process of counting.
This process may also be thought of in a graphical, visual way: for any set of numbers that can be placed on a number line, addition of two numbers may be defined as the process of moving a distance from the origin equal to the first number, and then continuing from that spot an additional distance corresponding to the second number.
Addition on the integers, rational numbers, real numbers, and complex numbers is defined by extending the concept of addition of positive integers to each type of number. For addition to be well-defined on any set (of quantities) it is necessary that any two elements of the set have a sum that is also an element of the set.
Addition may also be defined on quantities that are not numbers (scalars). Addition of vectors is defined by pair-wise addition of the corresponding components. Other kinds of addition, such as concatenation of strings of symbols, may also be defined.
Addition of ordinary numbers is associative and commutative, which means that the sum is the same regardless of the order in which addition is applied to the addends. In the case of some kinds of abstract quantities operations of addition may be defined that fail to be associative, or commutative, or both.