I read a cool book called "The Joy of X" by Stephen Strogatz in which he presents prime numbers in a way I like. Think of a prime number as a number than can be expressed as a square or rectangle no smaller than 2 by 2 with no left overs.
By this definition, it is easy to draw prime and composite numbers and it makes it easy for students to see why one is easily factored while the other cannot.
If you look at this first picture you will notice that it is not until you get to four that you have a 2 by 2 shape - a square. This means that 2 and 3 are prime numbers because they do not form the right shape.
Now look at the following illustrations for 5, 6, and 7.
Notice that the drawings of composite numbers also illustrate at least one set of factors. The factors of 4 are 2 because 2 x 2 is 4 or 4, 1 because 4 x 1 is 4. Where as the 5 shows no nice factoring and that makes it easy to tell it is a prime. For larger numbers, this is where the rules of divisibility come in or factoring trees.
I really like the way Stephen explained it because it gave me a way to help my students "see" a prime number as more than a definition.