These numbers are real in contrast to complex numbers which contain an element which is called imaginary.
All this arises because the square of any "normal" number is non-negative. So it is not possible to take the square root of a negative number. In some cases, though, there are mathematical advantages in being able to do so. So mathematicians introduce i, the square root of -1. Numbers involving i are said to be imaginary and, by way of contrast, those that don't, are said to be real. A number comprising a real part and an imaginary part is called a complex number.
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Yes, all natural numbers are real numbers. Natural numbers are a subset of real numbers, so not all real numbers are natural numbers.
All rational numbers are real numbers.
Complex numbers are a proper superset of real numbers. That is to say, real numbers are a proper subset of complex numbers.
Real Numbers cannot be the square root of a negative number. Real Numbers are not divided by zero. Basically, Real Numbers cannot be anything that is undefined.
Yes, -1 and 1 are real numbers. Real numbers consist of irrational numbers, rational numbers and integers.