una taza = a cup (as in a china or ceramic cup or mug with a handle)
un vaso = a glass (but also a paper cup
una copa = a glass (more like a wine glass)
The difference is down to the shape, not the material. A 'taza' always has a handle but a 'vaso' never does. A 'copa' generally refers to a wine glass type shape with a stem - something more elegant than a 'vaso' which is more straight sided like a tumbler. 'Copa' is also used for 'cup' in the sense of a trophy for sports etc. 'World Cup' = 'Copa del Mundo'.
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