There are several kinds. Natural numbers are finite cardinal numbers. Integers are whole numbers, including negative numbers.
Rational numbers are ratios of integers, sometimes called 'fractions'.
Complex numbers include so-called 'imaginary numbers', which are square roots of negative real numbers. Arithmetic, number theory, and real and complex analysis study the structures of the various number systems. There are philosophical problems concerning the ontological status of the various numbers - do they exist, are they mental, etc. There are theories of infinitely large numbers. Contemporary set theory , derived from the work of Georg Cantor and Ernst Zermelo, studies both infinite cardinals and infinite ordinals.