What is a sword with a curved blade called?

Entirely depends on the nation and the type of sword.

the Japanese curved blades are called wakizachi, katana, and no-dachi respectively for a short, normal length, and extra long version respectively.

A chinese curved sword is commonly called the dao

on the indian subcontinent, the Talwar is curved, while in afghanistan, the Pulwar is of the same basic shape.

In the middle east, the shamshir, also commonly called a scimitar was curved, as were the kilij and saif. It may be noted at this point that the image of the "curved arabic scimitar vs the straight knightly sword" during the Crusades is in fact myth; the arabic swords of the medieval age were in fact straight. The first curved islamic swords appear among the mamluks in the 14th century, long after the crusades ended.

In eastern europe, the hungarian or Avar sabre was commonly curved.

in western europe, many falchion were likewise curved swords, as were the later cavalry sabre and cutlass.

(A number of african and indonesian sword types also existed with curved forms, but I'm afraid I cant give names for them.)

Original answer:

A samurai sword