It is an integer, and hence it is a rational and real number as well.
This venn diagram from this site shows the different sets of numbers. You can see that if a number is part of a smaller set, then it is part of the larger sets around it.
Whole numbers are the same as natural numbers. They are numbers like 1, 2, 3, which are used for counting. -2 is not one of these.
Integers are numbers that do not have a fractional part. That is, if you were to write them as decimals, there would be nothing but 0 after the decimal. -2 is an integer.
Rational numbers are numbers that can be expressed as a fraction. The number -2 can be expressed as ##-2/1## and so it is a rational number.
Real algebraic numbers are those which are the zero (or root) of some polynomial function. -2 is a root of the function ##f(x) = x+2## and so is an algebraic number.
Real numbers are those that do not have an imaginary component. The imaginary unit i is defined as the square root of negative one: ##i= sqrt(-1)##. Since -2 does not have that, it is real.