An Ansatz is a guessed solution, or general form of a solution, to a mathematical problem.
A mathematician making the Ansatz will then attempt to prove that the Ansatz is indeed a solution to the original problem.
If he or she succeeds, the Ansatz is justified retroactively.
The word Ansatz is German, and means approach, or attempt, or beginning.
In all cases, the implication is that this is a first tentative step towards finding a solution.
Also note that, when used in this context, the word always starts with an uppercase A.