A prime number is a positive integer grater than one, that can only be exactly divided by the positie integers 1 and itself without leaving a remainder. For example 7 is prime, because it can only be divided by 1 and 7. If you divide it by say 2, you get 3 with a remainder of 1. Because you get the remainder, the division is not exact.
So a prime is a whole number, and the only whole numbers that exactly divide into it, without a remainder, it are one and the number itself.
Euclid proved thousands of years ago that there are an infinite number of prime numbers. They tend to be spaced fairly close together when they are small numbers, and get further apart as they get bigger. As of now, the largest known prime has over 17 million digits.
One major use of prime numbers is in the field of encryption. Sometimes web pages contain personal information and information that you want to keep strictly private such as credit card numbers. You would not want anyone 'listening in' on the Internet to see this information.
To do this, the pages can be encrypted during transmission (turned into gibberish using a secret code). The coding method uses a technique involving a large number that is the product of two prime numbers. It turns out to be extremely difficult to find these factors unless you know them in advance.
Composite numbers are the opposite of prime numbers. They are integers that do have integer factors. Any integer that is not prime is a composite number. It is 'composed' of other integer factors.
For example 12 is a composite number (and so not prime) because it has the integer factors 2,2,3 :