Which planet has the most moons?

At the moment Saturn has the most moons of any planet. The count stands at 82 for Saturn and 79 for Jupiter. (Last upated: November 2020)
Uranus as 27 moons, Neptune has 14 moons, Mars has 2 moons, Earth has 1 moon, Venus and Mercury have no moons.

What makes a moon a moon?

In the past this was a reasonably easy question to answer. A moon is a natural (not man made) object orbiting a much larger object that is not a Sun. However this question is becoming more and more difficult to answer because we can see that some planets, like Saturn, have an enormous number of objects orbiting them ranging from grains of dust to metre wide boulders and all the way up to small planet sized moons.

Even the International Astronomical Union doesn't have a detailed definition of what a moon is and only produced a definition of what a planet is in 2006. So it's clear that the count of the number of moons of these planets is dependent upon how the scientists choose to define the objects they detect, and not only how many objects have been detected.

How accurately are the number of moons known?

It is almost certain that there are more objects that will be classified as moons still to be found. It is also known that some moons have been lost in that they were detected many years ago, but that they have not been seen recently. It must be understood that many moons are very small and are very difficult to see from Earth. The Jovian and Saturnian systems are very complex and very small inaccuracies in determining a moons orbit can mean it's actual location may be wildly off from where one might expect it to be. It's unclear when lost moons might be struck off the list.