A prime number (or a prime) is an element of the natural-numbers greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number ... The fundamental theorem of arithmetic establishes the central role of primes in number-theory:
Any integer greater than 1 can be expressed as a product of primes that is unique up to ordering.
Here you get the first 50 millions of primes.
The concept of prime numbers is extended in ring theory, where an element $p$ of a ring $R$ is prime if and only if whenever $p\mid ab$, then $p\mid a$ or $p\mid b$.
One can easily see that this extends the definition of prime numbers in the natural numbers.