Primary cilium consists of the basal body (mother centriole anchored to the plasma membrane via set of appendages at its distal end), the transition zone (a specialized compartment implicated in targeting and sorting proteins in/out of cilium), and the axoneme (a membrane enclosed, microtubule-based structure protruding from the cell out).

Primary cilium is often viewed as cell "antenna". Indeed, it contains  receptors and effectors of several signaling pathways (the best known example is the Hedgehog pathway), so its is no surprise it is important  for modulation of signal transduction from outside to the inside of the cell. 

Pathophysiologies related to structural/functional defects in cilia are termed ciliopathies.

Centrioles are microtubule-based cylindrical structures.  In proliferating cells, centrioles duplicate exactly once per cell cycle - through the formation of one new centriole close to the proximal end of each pre-existing centriole.  

As a cells approaches mitosis, the centrioles mature (e.g. acquire  distal and sub-distal appendages), recruit pericentriolar material and form the centrosome. After the cell division, the older, fully matured ("mother") centriole of the centriole pair  can serve as basal body to trigger ciliogenesis.

Pathophysiologies related to centrosome anomalies are for exemple cancer,  primary microcephaly, and primordial dwarfism.