A flash meter is an exposure meter dedicated to measuring flash output. Early exposure meters were very simple and could only measure ambient light. The electronics necessary to measure flash were then bulky and expensive, so dedicated flash meters were made (e.g. Gossen Sixtron). Unlike an ambient meter, a flash meter must first capture the brief output and then display its intensity, which requires electronics.
With the reduction in both cost and size of electronics, more recent exposure meters are electronic in any case and often incorporate both ambient and flash measurement.
Flash measurement requires a fast reacting light sensor, such as the so-called Silicon Blue Cell (SBC, a silicon photo diode). Combined ambient & flash meters usually have two cells, one for each type of light measurement.