Prime numbers are positive whole numbers that can be evenly divided by exactly two numbers - 1 and the number itself. For example, 7 is a prime number because it can be evenly divided only by 1 and 7. Go up one number to 8, and that isn't a prime number because it can be evenly divided by 1, 2, 4, and 8.
There are an infinite number of prime numbers; here are the first ten: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29. A quick Internet search on list of prime numbers will turn up much longer lists.
Some things to note: Notice that the number 1 is not prime; this is because a prime number must be divisible by two distinct numbers, and 1 is evenly divisible only by 1. Also, because all even numbers are divisible by 2, 2 is the only even prime number. All other prime numbers are odd.
There is also a special set of prime numbers called Mersenne primes, named after French mathematician Marin Mersenne. A Mersenne prime is a prime number that is one less than a power of 2. For example,
25 = 32
32 - 1 = 31
Because 31 is a prime number and 32 is a power of 2, 31 is a Mersenne prime. Mersenne primes are very rare, and no one knows for sure whether there are an infinite number of them. One group calling themselves GIMPS (The Great Internet Mersenne Prime Search) recently discovered the 44th Mersenne prime: 232,582,657 - 1. This new Mersenne prime has a whopping 9,808,358 digits, 650,000 more than the 43rd Mersenne prime!