What does Pvu I, Pst I, EcoRI,Cla I, Hind III,BamHI, Sal I,Pvu II and rop in the circular figure of cloning vector indicate? How were these names derived?Should we by-heart all these names? What are their functions? How do they get their locations in the diagram?

Pvu I, Pst I, EcoR I ....... all these are names given to the restriction sites on a particular cloning vector. These names are given based on which site is cut by which restriction enzyme like site named as Pst I is cut by the Pst I restriction enzyme and so the other are named. At this level you don't have to remember all of them.
The function of these sites is to provide a place for a restriction enzyme to cut, so that when we are cloning a vector we can easily insert a gene of our interest by cutting the vector at restriction site and making space for the new gene.