Your moon sign is where the Moon stood when you were born. Where the sun sign is the long arc of who you are becoming, the Moon is the private register: what you feel before you decide anything, and what it takes to settle you.
The sun sign comes from the date alone and describes the long direction of a life. The rising sign is the surface: what people meet in the first ten minutes. The moon sign is the one nobody sees from outside and everybody lives from the inside — the reaction before the thought, the thing that has to be true before you can rest. It is also the placement people recognise fastest when they finally read it, because it names something they had assumed was just their private temperament.
Often you don't — but far more often than people assume, you do. The Moon spends about two and a quarter days in a sign, so on most dates any hour gives the same answer. It crosses on 44 dates out of a hundred, though, and on those the hour is the whole answer. This page checks your date and tells you which of the two cases you are in.
Almost always the crossing day. A calculator that asks only for the date has to assume a time, and most silently assume noon; if you were born on a crossing day and on the other side of it, their answer is simply the wrong one of the two.
Emotional instinct and what makes you feel safe: the unconsidered reaction, the way you comfort yourself, what a bad day sends you looking for. Traditionally it is also read as the relationship with home and with whoever raised you.
They answer different questions, so neither replaces the other. People who feel their sun sign describes a stranger usually recognise themselves in the Moon immediately — that is a good sign that both were needed rather than one.