Also known in England as a DHC (Drop Head Coupe). Strictly speaking, a convertible has roll-up windows and a padded fabric tops with head liners that hide the frames. The top can be lowered but not removed from the car. This makes a convertible much more weather tight than a roadster (aka OTS), which has a completely removable top.
It's interesting to note that early Jaguars such as the XK-120 were sold in both roadster and convertible configurations.