Since Darwin's theory of natural selection, and Pasteur's demonstration that abiogenesis is not a trivial matter the question as to how and where life first originated has been of great interest. Theories abound but practical demonstrations are notably lacking.
In this post I shall use the following definition of life:
1) It can reproduce.
2) It has variation.
3) The variation has no bound.
4) The variation is inherited.
These four characteristics are what is necessary for Darwinian selection to act.
On our current understanding of the tree of life there are three domains: Archea (heat loving prokaryotes...), Eukaryotes (plants, animals, fungi...) and Bacteria (note that there are two other models with significant support). These three domains seem to share a common ancestor in the distant past (as bacteria can exchange genetic information there may have been more than one). This ancestor would have been a single celled organism perhaps much like todays bacteria.
But bacteria, as they survive today at least, are far too complicated entities to arise purely by chance. So some simpler thing must have preceded them with the capacity to evolve into bacteria. In the case of bacteria this would also be a living entity but the arguement goes that at some point going back along the chain of ancestors we must reach an entity that does not have a living ancestor but arose through other processes. Then the task of explaining the origin of life is to explain how that first entity arose.
Here is some salient information that can guide us at least some of the way towards the answer:
A) Many of the basic building blocks of life can be produced with chemistry that occured on the early earth. Organic chemicals have been found in deep space, on comets and may have arrived on earth during the early bombardment of the earth soon after its formation.
The chemical constituents of life were around on the early earth. There seems to be little reason to believe that life arrived from elsewhere (transpermia).
B) DNA requires RNA and several proteins to replicate itself.
Life almost certainly didn't start with DNA as the likelyhood of having these proteins and RNA strands lying around is very small without a process like evolution to explain them.
C) A Ribozome is the machinery in cells that makes proteins (by translating mRNA) from amino acids. The Ribozome is composed of a type of RNA called tRNA (together with a few proteins which are known to be unecessary). The enzyme that transcribes DNA from RNA is made of single stranded RNA itself.
RNA can act as an enzyme and as a store of genetic information. Life probably passed through an RNA only stage where RNA played a jack of all trades role. Only later did DNA surplant RNA as a better way to protect that genetic information from copying errors.
So a popular theory of the origin of life is the RNA world hypothesis where RNA started the ball rolling and evolution took over from there first generating robust cellular machinery (a task that took over a billion years), then generating multicellular life, brains and so on.
Personally I think it holds a lot of truth in it. But there are still difficulties for this hypothesis to overcome. RNA is difficult to synthesise in the lab. We have yet to identify the key chemical steps in its formation. RNA is not stable under UV radiation (prevalent in the early solar system) and finally RNA does not have sufficient replicating fidelity to replicate accurately when the strands are longer than around 100 base pairs. The smallest known RNA enzyme for replicating RNA accurately is more than 200 base pairs long.
I'd also like to put down a few thoughts I've had into the question:
1) The definition of life given above is a definition relative to the conditions prevalent. The first forms of life might be wholly incapable of surviving today. The earliest forms of life may have lived in a different medium to that which modern life forms lives in.
2) The first forms of life could have been very bad at the task of living so long as they were capable of replicating within their life span. I think we tend to assume that the first replicator must have been very efficient and accurate.
3) The origin of life is often considered in terms very specific to life on earth. We assume that it has something to do with carbon, energy, polypeptides, entropy and 3 dimensional space. It is not at all obvious that any of these things are necessary. I think a good theory of the origin of life should work in a broad class of possible universes. An implication of this viewpoint is that the origin of life is a highly probable event. If it doesn't happen in one time or place then it would happen in another and if not then maybe we are looking at the universe at the wrong scale. Chemical reactions can occur at the timescale of a femtosecond (10^-15) so that gives you an idea of how slow everyday life is! If an origin of life seems unlikely on one timescale why should it be unlikely at another?
There is a lot of talk about our universe being fine tuned for life. If an origin of life is to be expected in a broad class of possible universes then this recent speculation would be very wide of the mark. If there really is something very special about our universe in terms of the development of life that would be interesting but I would expect life to be an emergent property of all sufficiently complex systems.
I'm guessing you'd like reasons for my intuition so here they are: Firstly the tendency for chemical systems to move towards stable configurations, the tendency for life to evolve to fit its surroundings and the tendency of life to adjust its surroundings to suit itself seem to me to be part of some general overarching theory of chemical and biological evolution that we are currently missing. Secondly the property of being alive (as I've defined it) is a definition that makes sense in practically any universe with a notion of time. So viewing life as a special property of carbon within our universe may well just reflect a lack of sufficient imagination on our part.