Do you mean exactly one letter goes into each envelope?
You have 5 choices which letter to put into the first envelope, then 4 choices which to put in the second envelope, then 3, then 2, then 1. 5*4*3*2*1= 5!
This is exactly the same as the number of different ways to order 5 objects.
Plato's answer was assuming that each letter had a correct envelope and you were asking "how many ways can you put 5 letters in 5 evelopes if no letter goes into the correct envelope". I did not interpret your question that way.
Here is a very helpful website for derangements
Here is a formula: .
Here is a good approximation: .
, so