Every proper variety X over a discretely valued field K has a model over the ring of integers O_K. The fiber over the closed point of O_K contains a lot of information about the variety X itself. This is why models are so often studied: the combinatorial data of the closed fiber are easier to handle.
I will talk about the two major results on models of K3 surfaces: the famous theorem by Kulikov and the undervalued Crauder-Morrison classification. I will of course introduce/recall the definition of a model of a variety and give some properties first.