The turning of the tassel is symbolic of graduation. It is a simple, clearly visible indicator of whether a student has been awarded his degree.
Like many quirks of academic ceremony, the turning of the tassel does not have a practical origin. The side on which the tassel hangs on a cap indicates a useful fact about the student wearing it, but the use of the tassel to display graduation status is rooted in nothing more than arbitrary tradition.
Tassel turning not only indicates completion of an undergraduate degree but is also a tradition at convocations awarding postgraduate degrees. Furthermore, the turning of the tassel is not always from the right side to the left, although this is most common. Some universities' students turn their tassels from the left to the right, and this is especially common amongst doctorate recipients.
The turning of the tassel seems out-of-place today because the context in which the tradition arose no longer exists. Historically, the academic dress of a gown and a cap (known as a mortarboard) was everyday attire for all university students. Today, it is seldom worn outside of graduation ceremonies, and so the turning of the tassels has little relevance or practical value.
Mortarboard tassels could traditionally indicate more than graduation status. The color of a cap or its tassels could tell others about a student's field of study, level of achievement or university of origin. Tassels were even occasionally customised for specific families in academia.