2: Biopolymer "Backbone" - AKA, why carbon?
We all know that earth life is carbon based, and science fiction likes to fantasize about 'non-carbon-based' life. Let us start by what it means to be a 'carbon based' life form. Simply put, nearly every non-solvent chemical that composes our body contains carbon in some fashion. It is not entirely incorrect to state that life as we know it is nothing but the consequences of the chemistry of carbon.
Why Carbon? We all know that the Sun is a mass of incandescent gas, a gigantic nuclear furnace, where hydrogen is built into helium at a temperature of millions of degrees, but this has further consequences when we consider the chemistry available to us right here on earth. To distribute heavy elements throughout the galaxy, they must first be built from the primordial elements by stars. Big Bang processes produce H, Deuterium, He-3, He-4, and Li-7. All stars produce He-4 from the three lighter isotopes, primarily through the chain of H + H -> D, D+H -> He-3, He-3+He-3 -> He-4 +2 H. Stars of sufficient size then further produce heavier elements by the fusion of He-4 - Because He4 is used as the primary building block, the majority of atoms produced from this process are simple multiples of He-4, mostly Carbon(3 x He) and Oxygen( 4x He ). These, along with other elements, are distributed throughout space via supernova. Because of the nuclear physics involved, then, the most common elements in the universe, are, in order, Hydrogen, Helium, Oxygen, Carbon, Neon, Nitrogen.
Of these six, Helium and Neon are noble gases, and do not react chemically under any reasonable condition, and are therefore uninteresting for our purposes. Carbon, however, is unique among these six elements in its ability to self-chain. With its four available valence electrons, Carbon is readily able to bind to itself, and is able to organize into both chains and rings, while retaining "free" bonds that are able to anchor atomic structures to the side of those rings, while avoiding crystallization. It is further able to maintain this 'chain' even while satisfying the valence requirements of hydrogen, oxygen, and other complexes. Furthermore, it does this in an approximate temperature band that includes that in which water is liquid, meaning that the three most common chemically reactive elements in the universe are able to, together, create a diverse and extremely complex chemistry, under readily obtainable conditions. When it comes to the chemistry of life, carbon gets there first with the most.
What, then, can replace it?
Most commonly proposed is silicon, however, there are certain issues with silicon. It is not thermodynamically advantageous for silicon to create multiple covalent bonds to another atom; therefore the preferred structure for silicon is crystalline. It does not react well with the broad variety of atoms and complexes required to sustain metabolism as we understand it, and silanes, long-chain Silicon/hydrogen molecules that are analogues to the simplest hydrocarbon chains, are unstable and spontaneously, and violently, decompose in water. Silicones, containing oxygen, however, are sufficiently stable, though large molecules are still less stable than their carbon counterparts. Further, silicon is generally unsuitable in a respiratory sense -- While at earth standard atmosphere carbon dioxide is a gas, we have a much more common name for silicon dioxide: Sand. Sand is very difficult to exhale. It is perhaps notable that the Earth contains more silicon than carbon, by three orders of magnitude, and yet Earth life is carbon based, not silicon. Some environments may exist where silicon is more valid than carbon for the chemistry of life.
Nitrogen - Phosphorus are also, together, theoretically, capable of performing the basic chaining functions of carbon in a life form. However, the high thermodynamic stability of the nitrogen-nitrogen bond, combined with the relative scarcity of phosphorus, significantly reduces the probability of such a life form. If this were to form, however, it would almost certainly require a methane and ammonium atmosphere, as noted above; for the metabolic gears to turn in the same direction as they do on earth, a nitrogen dioxide atmosphere would be required, and this would spontaneously decay into an N2-O2 atmosphere due to solar radiation.
Boron - Boron would suffice, and possibly even create a much wider availability of chemical complexes for the Boron based life form to work with. Unfortunately, it is comparatively rare - there simply is unlikely to be enough of it anywhere to form the basis of life. Furthermore, the boron oxides are all solids.
Other combinations, including metallic oxides, are theorized, but again, unlikely except in extreme conditions.
So far as we can determine, carbon is the primary candidate for the formation of life almost anywhere we would care to go.