Altium Generic components
There are two different systems for reffering to the size of SMT passives, confusingly these two systems use the same code for very different sizes.

The imperial codes are the dimensions in 10s of mils rounded to the nearest whole number (except in the case of 01005). The metric codes are the dimensions in tenths of a millimeter again rounded to the nearest whole number.

In my experiance the imperial system is more common. 

"0603(1608 Metric)" is giving the size code in the imperial system first and then giving the equivilent metric size code in brackets.

If someone just says 0603 then they probablly mean "0603(1608 Metric)" but it's possible they mean "0201(0603 metric)".