On LC centers: this is probably obvious for most people, but I suppose the point of an LC-center is that that's why a pair is not KLT. They are also called "non-klt" centers. I believe Christopher Hacon prefers this terminology. As far as I understand, LC centers are similar in spirit to associated primes. The simplest thing about a  module is its support and in particular its irreducible components. However, considering associated primes gives a better understanding as there are components that kind of should be considered part of the support, but they are embedded, sort of "shadowed" by other components. The union of LC centers is exactly the non-klt locus, but just like with associated primes, there may be some that are embedded, so knowing the LC centers gives more refined information about the failure of the pair to be klt.