If one wishes to get a clear image about what is a quantum object, it is very recommended to read the Feynman's path integral,
The electron behaves as a wave because inside an atom, it behaves as a quantum object, not as a classical object. A classical body has a trajectory. But a quantum object has a wave, a wave-length, produces interference as a wave. In the atom, the movement of the electron doesn't fit the classical laws of movement, i.e. rotates with some velocity around the nucleus. It is even difficult for us to say if it rotates. The best description we can give for the dynamics of the electron in the atom, is the wave-function.
Why is it difficult to say that it rotates? In the classical mechanics, an object that rotates around a rotation axis has an angular momentum along that axis. And in the plane perpendicular to that axis, the projection of the angular momentum is zero. But, for instance, in a hydrogen atom the electron on the lowest energy level has angular momentum zero. So, does it rotate? Hard to say.