"a number" is not a well-defined term.
But we'll start with the integers. The usual way to define the integers if you need such a formal definition is:
Let 0 be the cardinality of the empty set, 1 the card of the set containing the empty set, now use the axiom of infinity in your set theory to construct the naturals.
the negatives are constructed by adding formal inverses (-n is be definition the number such the n+(-n)=0
what we have now is a ring, in fact a domain, so we may localize with respect to the non-zero divisors, ie all non-zero numbers, and the field of fractions so formed is the rationals.
this field has a metric on it, a distance. it is not complete with respect to that metric. the completion is the real numbers. elements of the reals that are not rational, ie are not equivalent to a cauchy sequence q_i = q for all i, q rational are called irrational.
a transcendental number is a real (or complex) number that does not satisfy a (finite degree) polynomial with integer coefficients (strictly speaking all polynomials must have finite degree).
infinitesimals are an extension of the reals obtained by adjoining elements called things like e (epsilon), which is, formally, less than all positive reals by an extension of the ordering on R. They are not used much in mathematics.
hyperreals similar to infinitesimals, in that they are a nice trick but not much used. plenty of websites will give you an introduction.
a set is an element in a set theory, or at least that is the modern view point. loosely they are collections of objects satisfying certain axioms
element means element in the usual sense of the word
as does array
and constant,
and variable.
