Nautical miles measure distance. 1 nautical mile is the angular distance of 1 minute of arc on the earth's surface. As these differ slightly (6108' at pole c.f. 6046' at equator) 6080 was adopted (this being it's approximate value in the English Channel). The International nautical mile is 1852 metres, so is very slightly different from the UK nautical mile.
The metre is a unit of length in the metric system, and is the base unit of length in the International System of Units (SI).
As the base unit of length in the SI and other m.k.s. systems (based around metres, kilograms and seconds) the metre is used to help derive other units of measurement such as the newton, for force.