jakebeal's answer is very good, but the Wikipedia article on DNA alternatives is also very thorough. In particular, when astrobiologists debate whether alien life is likely to be carbon based, the answer is often "yes", because no other element has the chemical flexibility of carbon. While silicon is in the same group and can form long-chain molecules, it simply lacks the diversity of total molecular options, and the chains it can produce are much shorter on average than carbon (because it's a much larger atom and makes weaker double-bonds). Note that any carbon replacement suffers the same problem as silicon: much larger and less chemically flexible.
Note that your article simply assumes a carbon-based backbone, and asks what other molecules could replace the common nucleic acids.
Some folks talk about the possibility of radically different life forms, such as crystals. Unfortunately, crystals are good at growing, but not good at adapting. It is hard to imagine any kind of metabolism in a crystal-based system.
Way out in the woo-woo space, non-scientific folks like to play with the idea of "energy-based life forms". While I fully confess to a failure of imagination here, I would just like to point out that all the life we know about works hard to maintain homeostasis, which gets increasingly difficult in high-energy environments. This is because it takes energy to maintain the low-entropy configurations of living bodies, and all systems must emit heat to do work. Of course, dumping heat gets increasingly difficult the hotter one's environment.
The most literal way to describe an "energy-based being" is one that has no normal matter as part of its body. A being made entirely of photons doesn't make any sense at all. That's because most photons simply don't interact with each other, so it would be like having a body in which none of the cells could communicate. Planck-scale photons could use pair production to "interact" with each other, in some kind of ephemeral body, but it is hard to imagine how this could remain coherent over any kind of timescale. Also, the electron/positron pairs would push the definition of "no normal matter".
Of course, we could just be talking about beings whose base metabolic rate is so high that they emit blackbody radiation in the visible spectrum (so that we see them as natural sources of light). They would be made of ordinary matter, but such beings would consume pretty significant quantities of energy relative to terrestrial life. We have a good example of a blackbody that emits strongly in the visible spectrum: the sun. It has a blackbody temperature of about 6000k, and requires a massive scale of thermonuclear fusion to maintain that temp. Needless to say, any beings operating in that energy regime would either be much, much larger than human-sized, or would require physics that is well beyond our current understanding (perhaps they are powered by hearts of micro-black holes?).
Obviously, such a biology would not naturally arise on earth. ;)