The Sygnus serves a dual purpose on Androva. It's a magical object, working like a key to unlock more complicated spells, and it's also a representation of the bloodline that the magician belongs to.
This is what Jax's Sygnus looks like. It's a seven pointed star, and it's silver, which is the colour of magical energy.
If you're an underage magician (like Jax and Darius), your Sygnus is locked until the coming of age ceremony, to prevent you from carrying out any dangerous magic. Until that time, the centre cannot spin. If you have read Stealing Magic, you will know that things change pretty quickly when Jax meets Shannon!