What is the relationship between Markov Random Fields and Conditional Random Fields?
Conditinal Random Fields (CRFs) are a special case of Markov Random Fields (MRFs).


  1.5.4 Conditional Random Field
  
  A Conditional Random Field (CRF) is a form of MRF that defines a posterior for
  variables x given data z, as with the hidden MRF above. Unlike the hidden MRF,
  however, the factorization into the data distribution P (x|z) and the prior P
  (x) is not made explicit [288]. This allows complex dependencies of x on z to
  be written directly in the posterior distribution, without the factorization
  being made explicit. (Given P (x|z), such factorizations always exist,
  however—infinitely many of them, in fact—so there is no suggestion that the CRF
  is more general than the hidden MRF, only that it may be more convenient to
  deal with.)


Source: Blake, Kohli and Rother: Markov random fields for vision and image processing. 2011.


  A conditional random field or CRF (Lafferty et al. 2001), sometimes a
  discriminative random field (Kumar and Hebert 2003), is just a version of an
  MRF where all the clique potentials are conditioned on input features:
  [...]
  
  The advantage of a CRF over an MRF is analogous to the advantage of a discriminative
  classifier over a generative classifier (see Section 8.6), namely, we don’t need to “waste resources”
  modeling things that we always observe. [...]
  
  The disadvantage of CRFs over MRFs is that they require labeled training data, and they are slower to train[...]


Source: Kevin P. Murphy: Machine Learning: A Probabilistic Perspective