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Second genesis is a very general and loose term that means a case of abiogenesis that didn't result in us, that is, life based around the molecule of DNA. It can refer to extraterrestrial life that has evolved on other planets, but in more specific cases it refers to life being produced on Earth in an independent event. It is also important to note that "second" may not necessarily be second chronologically, it is entirely possible that the life we see on the planet now originated from the second or third (etc.) forms of life.
The importance of discovering a second, independent abiogenesis is huge as it affects the probability of life spontaneously arising on other planets in the galaxy and universe. If a second form of life (sometimes termed "shadow life") is found on Earth, it means that the odds of life appearing elsewhere in the universe can be determined to be much higher. Such a project to search for life on Earth would be minuscule in cost compared to exploring space but would still give SETI a much needed confidence boost.
Not to mention the fact that such a discovery would be a very firm kick in the balls to creationism, which likes to protest that abiogenesis is extremely unlikely. Having it happen twice would be interesting - but of course, there is no evidence that it has happened at this time.
 Artificial life
There have been studies to produce DNA like molecules artificially in the lab. These self replicating molecules have been shown to undergo some form of evolution, but it is unlikely that they will produce anything complex in time for their lab coat wearing creators to see it. If life is defined as something that can undergo evolution through natural selection (transferring and conserving infomation from one generation to another), then these may well be the first artificial life produced and essentially a confirmed second genesis on the planet Earth.
It is important to note that despite the artificial nature of these molecules and the "intelligence" of the scientists making them, they were still produced through natural, and not supernatural, means.
 A "true" second genesis
Determining a true second genesis is difficult. Indeed, because abiogenesis is a process rather than a magical single event, it's possible that it's happening all the time but never gets far along enough to be noted. As is noted with the Peanut Butter Argument, a second form of life would be quite weak compared to the established first form, and would be very unlikely to survive. Defining what constitutes a "second" genesis also causes problems as the principles of natural selection show that isolation can very rapidly create new and very different species. If natural selection can occur at the chemical level - which is likely - it may well be that these hypothetical "second genesis" life forms do, in fact, share a common ancestor with DNA-based life, but this ancestor will be very old, predating the formation of cells and possibly the use of proteins.
The confusion of what would be a "true" second genesis was shown in December 2010 when leaks of a NASA press conference suggested a second genesis of arsenic utilising bacteria. The actual release was impressive, but not quite a "second genesis", and more of a very extreme adaptation where bacteria could replace the phosphorus in its DNA with arsenic.
This is also an issue for the search for extraterrestrial life, as there is a remote possibility that life that has evolved on other planets may share some kind of ancestry with life on Earth. This hypothesis is generally termed panspermia.
 See also
- Artificial life 'could be created within five years' This was published in 2009, will we get new life by 2014?