“Prions are infectious agents composed entirely of a protein material that can fold in multiple, structurally distinct ways, at least one of which is transmissible to other prion proteins, leading to disease in a manner that is epidemiologically comparable to the spread of viral infection. Prions composed of the prion protein (PrP) are believed to be the cause of transmissible spongiform encephalopathies (TSEs) among other diseases.”
Source : Wikpedia
Prions don't contain DNA, so aren't normally considered to be 'alive' [also see Defining Life]. Nevertheless, prions can readily replicate. The mechanism(s) by which they do so is still not yet determined.
There are currently two main theories : The 'heterodimer' model and the 'fibril growth and fibril breakage' model. Neither explains all the observed manifestations. See: Prion replication mechanism