The millimetre (American spelling millimeter) is a unit of length and is a multiple of the SI unit metre. It uses the symbol mm.
There are 1000 mm in 1 m and 10 mm in 1 cm. From this one can derive that there are 0.03937 in.
The millimetre is sometimes verbally abbreviated to 'mil' - which can cause confusion in the US as there is a unit there that means thousandth of an inch.
The em is a unit of length in the field of typography and uses the symbol em.
It is a "relative" unit and equals the currently specified point size. For example, 1 em in a 24-point typeface is equal to 24 points. This allows a typographer to specify text size relative to a base figure.
In metal type, the em was equal to the line height of the metal body. This meant the physical size of the letter could not exceed the em. In digital type, the em is a grid of arbitrary resolution so can be scaled to any point size for screen or print.