Abiogenesis is the technical term for the spontaneous formation of life from non-living matter. It is typically imagined to be a long sequence, with many parts that must come together to form a complete self-replicating system. It does not require that the life that is created use DNA, cells, or any of the hallmarks of Earth's life, only that life arise from non-life. Most abiogenesis studies work from either the bottom-up, seeking to create life in the lab, or from the top down, seeking to strip away all but the most basic functions of life.
The first experiments in abiogenesis during the 1950s by Stanley Miller were focused on Amino Acids, producing these molecules in the lab in what was thought to be a simulated early Earth atmosphere/ocean system. While these experiments were successful, later scientific discoveries disputed the active atmosphere used in the experiments in favor of a more stable atmosphere. New research into more exotic environments such as early icecaps and deep-sea vents has produced more recent hopeful findings.
The next stage in the evolution of proto-life would have been the creation of a polymer, perhaps one that would replicate itself. Here are two competing theories, the iron-sulfur metabolism first theory, and the RNA-world replication first theory. Iron-sulfur theory uses a primitive metabolism to generate products that fuel and or catalyze the production of new processes. It also introduces a primitive cellular membrane in the form of lipid bubbles to contain this process. RNA-world on the other hand has a cycle of replication at its root. One of the keys to this process is the formation of ribosymes from RNA, self-catalyzing its own production. As RNA replicated itself, the fastest replicating molecules would be the least likely to break down before they could replicate again. If a molecule happened to encode a separate product like a protein that would help its replication speed, this would also help it produce more 'offspring' and make it less likely to go 'extinct' One complication of this theory is the short half-life of most nucleic acid species. To avoid problems with both theories, some scientists are now combining them into one holistic origin of life's two most vital processes.
The final stage of abiogenesis was the formation of the first cells. The metabolic and replication processes would have been streamlined greatly, eliminating much of the molecular evidence of previous stages. DNA also comes into its own as the storage medium of choice for all life's replication needs.