It is widely held by biologists that the first self-replicating molecules, RNA perhaps, appeared on Earth about 4 billion years ago. How did this happen? Scientists don't know, but they are working on it. This is one of the unsolved problems that IDers adduce as evidence for intelligent design.
There are three ways to approach the problem without invoking miracles. One way is to synthesize RNA in the laboratory under conditions that might have existed on the early Earth. So far, this hasn't happened. Another way is to simulate with computers the molecular chemistry that might have given rise to RNA. A third way is to invent mechanical or electromechanical "organisms" that self-replicate -- and that might conceivably have got themselves together on their own from components with sufficient potential.
In the September 29 issue of Nature, scientists from the Center for Bits and Atoms at the MIT Media Laboratory describe a 5-unit-long electromechanical "molecule" that can assemble copies of itself from randomly distributed parts floating on an air table. Self-correction is built into the system.
You can watch a seed "molecule" reproduce itself here (and in the last frames get a sense of scale).
What does this prove? Nothing really, except that curiosity, inventiveness and perseverance can be both instructive and fun.