In math, idempotence describes only unary functions that you can call on their own output. Math-idempotence is, "If you take the absolute value of a number, and then you take the absolute value of that, the result doesn't change on the second (or subsequent) operations." Math.abs is math-idempotent. Math-idempotence only applies to functions of one parameter where the parameter type and return type are the same. Not so useful in programming.
Here's the real definition.
In computing, an idempotent operation is one that has no additional effect if it is called more than once with the same input parameters.Programming-idempotence is about side effects. It's about stuff that happens to the outside world when you call a function. Idempotence says "If you've called me once, it doesn't matter whether you called me again."
Not idempotent: Pushing "Place order" creates a new order in the database.
Idempotent: Pushing "Place order" moves order 4567 from 'cart' to 'finalized' status.
Not idempotent: Fetch a row from the database and create an object to represent it
Idempotent: Fetch a row from Hibernate, which returns the previously allocated object for the same row on a second call.
Idempotent: Give me that pizza.
Not idempotent: Make me a pizza.
Idempotent: check a password against the hash in the database
Not idempotent: check a password and increment the bad-login-attempts if it's wrong
Not idempotent: Install MongoDB
Idempotent: Install MongoDB if it isn't already installed
Idempotent: perform a calculation and return the result
Not strictly idempotent: perform a calculation, write to the log file, and return the result
Keep it idempotent
Programming-idempotence, every function has it or it doesn't. Data-in, data-out functions do not affect the outside world at all, so they are naturally idempotent. If your function does something to the outside world, consider what will happen when your function is called twice. Will it blow up? cause twice as much to happen? or have the same result as calling it once?
It's a good idea to make side-effecting functions idempotent. They make your forms immune to repeated-submission problems. They make your chef scripts re-runnable when a download fails in the first attempt. They avoid surprising people.
The word idempotence came from math, but its meaning in programming context is broader and more useful.
 "data-in, data-out" is a more accessible term for "Referentially transparent and nullipotent." Nullipotent means that the outside world doesn't care whether you called it 0 or 1 or 100 times.
 If you want a mapping from the math-idempotency to programming-idempotency, think of it this way:
All functions (in a language other than Haskell) have an invisible parameter: the world around them.
When we write function application in our program, we supply the declared parameters. The world around
is supplied at runtime.
Similarly for output, our program receives an explicit return value, and there is an invisible return
value as well: the world after our function exits.
Therefore, consider every function application in code as a partial application, with the world as the single
unsupplied parameter. Looked at this way, every function + inputs is a unary operator on the single
parameter World, and it returns another World. Calling our function again with the same inputs, does it change
that World again?
f(x) = f(f(x))
f(a,b)(world).newWorld = f(a,b)(f(a,b)(world)).newWorld
where the method newWorld extracts the world-after-execution from the function's multiple return values (the explicit one and the implied world).
 World is one of the most awkward words in the English language.