What is a metro?
My personal definition is:
1) An urban electric mass rail transport system, i.e. it is primarily used to move within the city
2) Totally independent from other traffic, rail or street traffic
3) High frequency service (maximum interval approx. 10 minutes during normal daytime service)
Obviously a metro does not have to be underground (this is why I prefer this term to 'subway' or 'underground' or 'U-Bahn' as all these terms imply a tunnel), it can also be elevated or at grade. A metro does not necessarily use heavy rail technology, therefore the Docklands Light Railway or the Lille VAL are full metro services, with the only difference that their capacity is smaller according to the needs of the city/area they serve. It's also irrelevant whether the metro runs on steel wheels or rubber tyres, is monorail or conventional double rail, uses third rail power supply or overhead wire, is fully automated or has a driver, has standard, narrow or broad gauge.
There are many mixed forms nowadays and probably there will be even more in the future. In Europe (especially Germany and the Benelux) light rail systems (Stadtbahn, Premetro) have been bridging the gap between metros and trams for some decades now and provide, in most cases, a perfect service in their city. For the time being I can only cover these mixed systems (which I'd like to refer to as metroTrams) in Europe only, although also in America and Asia more and more networks of this kind are being built.