Zero is an integer. An integer is defined as all positive and negative whole numbers and zero. Zero is also a whole number, a rational number and a real number, but it is not typically considered a natural number, nor is it an irrational number.
A natural number is a number used in counting. The set of natural numbers begins with one. A whole number is the set of natural numbers plus zero. Integers go a step further, by adding whole numbers with a negative sign preceding them. Rational numbers include all of the integers plus fractions and decimals. Real numbers include all of these classifications plus the set of irrational numbers. Irrational numbers are not integers or fractions, but are decimals that go on without a pattern.