In the moped world, rigid means a moped with no rear suspension. Most manufacturers made at least one model of moped with a rigid frame. Rigid models were often the "budget" models of mopeds when they were originally sold. Most rigid mopeds make up for the lack of rear suspension with a big, cushy seat. The most common rigid moped is probably the Vespa Ciao.
Rigid frames from all moped brands are highly sought after today due to their light weight.