In this paper, we propose a characterization for non-elementary trapping sets (NETSs) of low-density parity-check (LDPC) codes. The characterization is based on viewing an NETS as a hierarchy of embedded graphs starting from an ETS. The characterization corresponds to an efficient search algorithm that under certain conditions is exhaustive. As an application of the proposed characterization/search, we obtain lower and upper bounds on the stopping distance <inline-formula> <tex-math notation="LaTeX">s_{\min } </tex-math></inline-formula> of LDPC codes. We examine a large number of regular and irregular LDPC codes and demonstrate the efficiency and versatility of our technique in finding lower and upper bounds on, and in many cases the exact value of, <inline-formula> <tex-math notation="LaTeX">s_{\min } </tex-math></inline-formula>. Finding <inline-formula> <tex-math notation="LaTeX">s_{\min } </tex-math></inline-formula>, or establishing search-based lower or upper bounds, for many of the examined codes are out of the reach of any existing algorithm. For a constant degree distribution and range of search, the worst case computational complexity of the proposed search algorithms for finding NETSs and stopping sets is linear in the code's block length <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>. The average search complexity for stopping sets, however, is constant in <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula>, if the simple cycles required as input to the search algorithm are available.