One interesting set of example is Salem numbers. These are real algebraic integer  all of whose  conjugates has absolute value less than or equal to one and at least one of the conjugates lies on the unit circle.

http://en.wikipedia.org/wiki/Salem_number

Also if all its conjugates lies on the unit circle then it has to be a root of unity, its known as Kronecker Theorem.

http://mathoverflow.net/questions/33169/good-effective-version-of-kroneckers-theorem