Yes, there are.
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Rational numbers are numbers that can be written as a fraction. Irrational numbers cannot be expressed as a fraction.
There are more irrational numbers between any two rational numbers than there are rational numbers in total.
Infinitely many! There are an infinite number of rational numbers between any two irrational numbers (they will more than likely have very large numerators and denominators), and there are an infinite number of irrational numbers between any two rational numbers.
Actually there are more irrational numbers than rational numbers. Most square roots, cubic roots, etc. are irrational (not rational). For example, the square of any positive integer is either an integer or an irrational number. The numbers e and pi are both irrational. Most expressions that involve those numbers are also irrational.
Irrational numbers are real numbers because they are part of the number line.