What is 'K' in the Bounded Post Correspondence Problem?
Suppose your list is M=(bba,ab,bba,a) N=(bb,aa,bb,baa) Then its equivalent tile representation is as follows



Now your pcp problem is equivalent to finding a sequence of tiles(repetition allowed) such that concatenating all upper halves and concatenating all lower halves gives you same string.
K here is nothing but number of tiles that can be used

Edit : As Yuval Filmus subtly mentions in his comment, there are many other interpretations of K. In an another formulation K can also be the maximum length of concatenated string, in which case the bounded PCP problem is Pspace complete. But usually many books refer to K as being number of tiles that are allowed.Please refer to Sipser book for more.