Why do most organisms have negative supercoiled DNA?
One important notion in DNA topology is the linking number ($Lk$) which is the sum of right-handed twisting # and left-handed writhing # ($Tw+Wr$). Say the $Lk = x$ for relaxed DNA. If $Lk&lt;x$ then you have negative supercoiling while positive supercoiling happens when $Lk&gt;x$.

The higher the $Lk$ value the higher the helical energy of the DNA and the harder it is to separate it meaning the DNA is now more stable. In so-called "extreme" conditions (e.g. high acidity or high T˚) the organism must protect his DNA from degradation. Therefore those organisms require a more stable form of their DNA which can be achieve by positive-supercoiling and/or high GC content.

Now why do the vast majority of the organisms have negatively-supercoiled DNAs? If you let the DNA by itself, it will adopt an $Lk=x$, i.e. it will be in his "relaxed"-form. Negative supercoiling arose because of the presence of Topoisomerases (type I and II), which are proteins that cleave the DNA and release the helical tensions and which cumulatively provokes the DNA to be slightly underwinded compared to it's relax-form. Is this good for the organism, if living in non aggressive conditions? Yes it is because a decrease of the $Lk$ will make the DNA easier to separate. This means replication and transcription can occur more easily or in proper terms, they now require less energy to perform their functions.

Here an excellent reference on this subject as well as a Youtube video that explains those concepts using strings. The latter is really helpful as those concepts are easier to understand visually than in writing.

I hope this helps.