My son recently asked me "What is a programmer?" when I told him what I do at work all day.
While I satisfied him with my simple answer, I started to think about it somewhat more explicitly - what does it mean to program?
A program is a sequence of instructions for a turing machine to complete an action or algorithm.
A programmer is one who is able to understand and therefore create programs from an abstract action description or algorithm.
More explicitly,
A program is a sequence of instructions for a state machine to perform:
In some small part almost every program performs all of the above, and a typical programmer is able to understand the elements of a program that allow each of those missions for the machine.
I've asked a similar question in a slightly different way: http://stackoverflow.com/questions/486028/what-exactly-is-programming-closed. Perhaps some of those answers will help.
Programmers give explicit instructions to extremely fast. powerful calculating machines that think we are perfect and never make mistakes. They will do exactly what we tell them to even if it is wrong, stupid, dangerous, or all of the above. They will also do it very quickly. We then spend most of time trying to find out what wrong instructions we gave them.
Young children I would simply say that I work with computers. However at a shockingly young age children start to become computer literate. They then will understand what we do.