The term "abiogenesis" comes from Greek roots: the prefix "a-" (without), "bio" (life), and "genesis" (beginning) — i.e., the origin of life from non-life. In this broadest sense, abiogenesis can include religious creation myths. More commonly, however, it refers to the scientific study of life's origins.
Abiogenesis and evolution
While evolution explains how self-replicating entities change over time, abiogenesis is the study of how self-replicators arose in the first place, and how evolution got started. It is thus related to evolution, but distinct.
Creationists often claim that since scientists do not know how life started, that this makes the theory of evolution invalid or baseless. This is untrue: just as it is not necessary to know the history of the internal combustion engine to understand how an automobile works, evolutionary biologists can study how populations evolve today without necessarily knowing how life arose.
Abiogenesis and spontaneous generation
Creationists often claim that Louis Pasteur disproved abiogenesis by showing that flies do not appear on meat if the meat is sealed from outside contamination. This idea often manifests in the so-called Peanut Butter argument against evolution.
In fact, Pasteur disproved the theory of spontaneous generation, the idea that fully-formed modern flies spontaneously arise from meat.
Current models of abiogenesis make no such claim. Rather, researchers try to figure out which organic molecules could have been formed under the conditions of the early Earth, how they might have combined to form RNA, DNA, cell membranes, metabolism, etc.
A multitude of hypotheses
Since evidence indicates life arose about 3.7 billion years ago, it is very difficult to find fossil remains of the earliest life forms. Many chemicals thought to have played a part in the origin of life do not last long under the conditions which they may be found today. The rocks where they might otherwise be found might have been contaminated by geologic processes, and many of them may have been subducted into the Earth's mantle. Nonetheless, there are numerous hypotheses as to how life could have arisen.
Some researchers believe that life arose on the surface of the Earth, perhaps as an oily film on the surface of the ocean, or in calmer tidal pools; the surface is, after all, where most living things are found today. Others argue that the surface of the early earth was bombarded with ultraviolet rays that would have broken down organic molecules almost as soon as they were formed, and thus these molecules could not have accumulated in sufficient concentrations to permit interesting reactions to take place.
It is also possible that life arose deep under the ocean, protected from ultraviolet rays, around hydrothermal vents. These could have provided the energy, in the form of heat, necessary for chemicals to form and react with each other.
Some of the chemicals required for life may have fallen to Earth in meteorites. Many chemicals, including sugar and alcohol, can form in gas clouds in outer space, and may therefore have been part of the composition of the Earth from the very beginning of its formation. Others may have fallen to Earth later. Meteorites known as carbonaceous chondrites contain many types of organic molecules, even after falling through the atmosphere and crashing to Earth.
It is hypothesized that minerals, including clays, may have played a role in the origin of life: if certain amino acids become attached to a clay surface, the clay in effect holds them in place, allowing other amino acids to become attached to the original ones.
Other minerals have microscopic pores, which may have been filled with interacting molecules, thus in effect playing the role of a primitive cell wall. However, it is known that lipids can spontaneously form hollow spheres in water. Thus, it is not clear whether metabolism came before cell membranes or vice-versa.
It is possible that we will never know exactly how life arose on Earth, but it may be possible to come up with a handful of likely scenarios.
The Urey-Miller experiment
In 1951, Harold Urey and his graduate student Stanley Miller conducted a seminal experiment: they filled a glass vessel with water, methane, ammonia, and hydrogen, by which they hoped to model the ocean and early atmosphere of the Earth (note that today we have a different picture of the composition of the early atmosphere). Two electrodes in the vessel produced sparks, simulating lightning. The vessel was connected by a tube to a condenser and a second chamber, from which samples could be withdrawn and analyzed.
Within a few days, the water turned yellow and dark "muck" had appeared on the walls of the vessel. This turned out to contain glycine, an amino acid. Later on, several other amino acids and other organic molecules were found.
Later experiments not only confirmed the results of the Urey-Miller experiment itself, but expanded on it, showing that many organic molecules are easy to synthesize under a wide variety of conditions, including different atmospheric compositions and energy sources.
Miller and Urey did not create life in the lab, of course, nor does anyone claim that they did. They did, however, demonstrate that molecules essential to living beings can form naturally under likely conditions of early Earth.
One common creationist argument is that the Urey-Miller experiment only created a few of the amino acids used by life, not life itself. Another is that the gases used by Miller and Urey were different from those actually present on primordial Earth.
According to a paper published in Science in 2008, researchers were able to reanalyze the residues from one of the original experiments, and found several amino acids that instruments in the 1950s were not sensitive enough to detect. In other words, Miller and Urey were more successful than they realized.
The paper also argues that the atmosphere used in that experiment may have been locally realistic. That is, that mixture of gases would not occur throughout the planet, but only near volcanic eruptions.
Later studies during the 1960s by Joan Oró, et al., that used atmospheric conditions that better match the actual (hypothesized) atmosphere of early Earth turned out to give even better results, turning up for example adenine, which is one of the nucleotide bases that form the "backbone" of DNA.