Cold fusion refers to a nuclear fusion process which occurs at or near room temperature, as compared to conventional nuclear fusion, which requires a very hot (100 million degrees) plasma. There is a number of such processes which are under investigation and are generally considered to be scientifically reputable, although none of them have reached anything close to breakeven, including muon-catalyzed fusion and bubble fusion. However, cold fusion is often used to refer to a claimed particular mechanism which is not considered viable by most scientists.
On March 23, 1989, Stanley Pons and Martin Fleischmann at the University of Utah claimed to measure a production of heat that could only be explained by a nuclear process. Steven Jones at Brigham Young University did not observe heat but claimed to observe neutron emission that would also indicate a nuclear process. The claims were particularly astounding given the simplicity of the equipment, just a pair of electrodes connected to a battery and immersed in a jar of heavy water. The immense beneficial implications of the Utah claims, if they were correct, and the ready availability of the required equipment, led scientists around the world to attempt to repeat the experiments within hours of the announcement.
This claim was surrounded by a lot of media attention and excitement which brought the phrase cold fusion into popular consciousness. A few months after the initial cold fusion claims, the Energy Research Advisory Board (part of the US Department of Energy) formed a special panel to investigate cold fusion and the scientists in the panel found the evidence for cold fusion to be unconvincing. 
The most common experiments involve a metal electrode (usually palladium or titanium) which has been specially treated so that it is saturated with deuterium and placed in an electrolytic heavy water solution. The experimenters saw extra heat coming from this system which was not readily explained by the electrolytic reaction itself. Although some experiments claimed to see fusion products (tritium, helium, or neutrons) the amount of detected fusion products did not match what was necessary to explain the amount of excess heat. The initial announcement by Pons and Fleischmann in March 1989 exhibited the discrepancy between heat and fusion products in sharp terms. Namely, the level of neutrons they claimed to observe was 109 times less than that required if their stated heat output were due to fusion.
The idea that palladium or titanium might catalyze fusion stems from the special ability of these metals to absorb large quantities of hydrogen (or deuterium), the hope being that deuterium atoms would be close enough together to induce fusion at ordinary temperatures. The special ability of palladium to absorb hydrogen was recognized in the nineteenth century. In the late nineteen-twenties, two German scientists, F. Paneth and K. Peters, reported the transformation of hydrogen into helium by spontaneous nuclear catalysis when hydrogen is absorbed by finely divided palladium at room temperature. These authors later acknowledged that the helium they measured was due to background from the air.
In 1927, Swedish scientist J. Tandberg claimed that he had fused hydrogen into helium in an electrolytic cell with palladium electrodes. On the basis of his work he applied for a Swedish patent for "a method to produce helium and useful reaction energy". After deuterium was discovered in 1932, Tandberg continued his experiments with heavy water. Due to Paneth and Peters' retraction, Tandberg's patent application was denied eventually.
In fact, even though palladium can store large amounts of deuterium, the deuterium atoms are still much too far apart for fusion to occur in normal theories. Actually, deuterium atoms are closer together in D2 gas molecules, which do not exhibit fusion. The closest deuterium-deuterium distance between deuterons in palladium is approximately 0.17 nanometers. This distance is large compared to the bond distance in D2 gas molecules of 0.074 nanometers.
There are still a few people trying to do cold fusion.
Robert L. Park (2000) gives a decent account of cold fusion and its history which represents the perspective of the mainstream scientific community.
See also: pathological science
Another table top candidate for fusion is through an extreme form of sonoluminescence, is often called bubble fusion. Natural bubbles of gas inside a liquid would be made to expand to near vacuum, and then collapse. The extreme pressures and temperatures needed for fusion could potentially be reached. Bubble fusion is often associated with cold fusion due to the use of small room temperature containers of acetone (although the fusion process itself would still take place under localised extreme thermonuclear temperatures and pressures). In 2002 bubble fusion attracted a significant amount of media coverage as controversial results were published. As the liquid researchers chose heavy acetone (acetone in which hydrogen atoms had been replaced by heavier deuterium atoms). It was hoped the deuterium atoms would be fused to form helium, releasing energy.
Unlike the cold fusion results of Pons and Fleishman, the bubble fusion results were published in a peer reviewed journal, Science. In July of that same year however researchers from the University of Illinois claimed they had discovered chemical reactions in the collapsing bubbles, sapping most of the energy available. Instead of a temperature of millions of degrees, they calculated the temperature within the collapsed bubbles would be closer to 20,000 degrees.