Both centrioles and asters are composed of the protein tubulin - tubulin is composed of two polypeptides, alpha-tubulin and beta-tubulin.   Tubulins assemble into microtubules. 

Microtubules generally form a radial array (an aster) centered on what is known as the microtubule organizing center  or centrosome.  

The term aster is normally used in the context of a dividing cell 

In animal cells, centrioles are located within the centrosomal region. 

Centrioles themselves are composed of one complete and two "incomplete" microtubules, organized as a "triplet" microtubule (there are also a number of other proteins, in addition to tubulin in a centriole). 

Nine triplet microtubules are organized in a ring, and there are often two centrioles (oriented perpendicular to one another) within the centrosomal region. 

When the cell is about to divide, these centrioles separate and two centrosomes migrate to opposite sides of the cell and become the poles of the mitotic spindle.