When applied to cars, 'Spyder' indicates that the car is a
convertible, supposedly because the framework of the soft-top resembles a
spiderweb. Presumably 'spider' is mis-spelled so that the first manufacturer to use the term could trademark it, although it is now in common usage.
It is typically used by Italian manufacturers. When used by manufacturers from other nations, it seems pretentious - with the exception of Porsche, whose Porsche 550 Spyder is widely regarded as one of Porsche's most beautiful cars (and, more perversely, it is famous for being the car in which James Dean lost his life).
Both Alfa Romeo and Fiat have used the name to good effect, although by far the most famous cars to be badged as spyders have been made by Ferrari. Spyder variants of the 365GT (the 'Ferrari Daytona', latterly of 'Miami Vice'*), the 355, and the 250 are some of the most evocative and sought-after cars ever made.
Perhaps because of the association with performance sports cars, 'Spyder' is also a popular internet nickname.
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* Although in that instance the car was a replica built on the chassis of a Chevrolet Corvette.