What does 999 on a silver coin mean?

It represents the "fineness", or purity, of the silver in the coin expressed as a decimal.

Normally fineness is given to 3 digits. In that case the number expresses the purity as a in parts per 1000. For example, 999 means 999/1000 of the coin is pure silver. The other 1/1000 is some kind of trace impurity that's impossible to remove.

Sometimes the fineness is expressed with a decimal point (.999) and sometimes without, but the meaning is always the same - e.g. 999/1000 = 0.999 = 99.9% (see why they made you take algebra in middle school?)

999 fine silver is generally only used for bullion coins and other items that aren't subject to wear, because it's too soft. In the US, pre-1965 silver coins minted for circulation were made of 900 fine silver, i.e. 90%, with the remaining 10% being copper for hardness. Sterling silver is 925 fine often used in jewelry and formal dinnerware.