The ribozyme is a piece of RNA which folds back on itself to make a 3-dimensional structure. This can act like a protein enzyme in catalyzing metabolic reactions in the cell (i.e. an enzyme makes reactions proceed millions of times faster than they normally would). However, a ribozyme does not typically catalyze the formation of copies of itself.
Ribozymes are appealing as the basis for an RNA-World hypothesis. The idea is that if these molecules, which carry information in the form of nucleotides like DNA, were able to catalyze more of themselves, then cellular life could begin to evolve. This system would be simpler than present living cells which need tons of proteins to copy the information contained in DNA and RNA.
Appealing as it sounds, the RNA-World hypothesis has a lot of problems such as how these fancy RNA molecules could spontaneously arise, the fact that self-catalysis only works for very short pieces of information, and that RNA breaks down very quickly. As a mechanism to move toward the living cell, this origin of life theory does not deliver a process that would work!