Yes, the Sun rotates. This can be observed by tracking a variety of features on the Sun, such as sunspots, X-ray brightpoints, coronal holes, filaments, and small magnetic flux elements. Another way to determine the rotational speed of the Sun is to measure spectral lines at the edge of the Sun's disk and determine their redshift.
It is thought that the rotation of the Sun is due to the way the primordial gas cloud collapsed in on itself to create the Sun. Also, it is likely that the Sun originally rotated much faster when it initially formed, than it does to day. This slow-down was probably caused by 'magnetic breaking' in which strong magnetic fields threading our primordial Sun out into the solar wind resisted the rotation.
Today, the reason that the Sun rotates at a different angular speed at different solar latitudes is due to hydromagnetic effects. One cause is thought to be the nature of convection within the outer third of the Sun. Interestingly, below a the convective envelope, beneath a boundary called the tachocline, the Sun rotates as a rigid body. You can find more about this in Schou et al., 1998
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