A prime factor is a factor of a number, and that factor is also prime. A prime number is one that cannot be neatly divided into by any other number than itself and 1. Factors are numbers that do wholly divide into another number. Here "wholly divide" means "leaves no remainder".
An example, then. The number 60 can be wholly divided by 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60. These are the factors of 60. The Prime factors are 2, 3 and 5, and 60 is 2235 ( used here to mean "times by").
From the prime factors we can find all the other factors by using all the possible combinations of prime factors, e.g. 22 gives 4 and 35 gives 15, and 415 = 60, while 23 = 6 and 2*5 = 10 gives another pair of factors. And so on.
Prime factors are useful for a number of reasons. Firstly, when working with awkward fractions it can be useful to know all the prime factors of the numbers invlved to help with simplifying the numbers. Suppose you had to work out (14/15) * (130/231). Doing this by hand could take a while and leave lots of chances to mess up. However if we write out all the numbers by prime factors we can find any possible simplfying before we do the sum.
In this example, we would get 14 = 27 and 130 = 2513, while 15 = 35 and 231 = 3711, so the sum could have been written (225713)/(335711). We see that 5 and 7 appear on top and bottom so we cross these out, leave (2213)/(3311), which is much easier to deal with and turns out to be 52/99.
Also, if you want to find out whether a number is prime, or if not what are its factors, you simply check whether any prime numbers divide into it. Take 887 as an example. You could test whether this is prime by checking every number below it to see if it divides into 887, but it's easier to just try 2, 3, 5, 7, 11, 13... the prime numbers, in other words, which reduces the amount of effort involved in discovering that 887 is indeed prime.