"Most common" means the forms most frequently encountered in the entire scope of chemistry, which I'm pretty sure corresponds loosely to the most thermodynamically stable oxidation state. It is not always the case that the '-ate' ion is the most stable, however—as an example, see the links for chlorine below).
A detailed answer necessitates noting that the nomenclature extends beyond just '-ate' and '-ite'. Most p-block elements that form oxo-anions form a whole series of them, each with the central atom in an oxidation state two away from each neighbor. For all elements except the lightest ($\ce{C}$ and $\ce{N}$) and heaviest ($\ce{As}$, $\ce{Se}$, $\ce{Te}$, etc.) , usually the series is considered to contain four members, though it is not guaranteed that all are stable or able to be characterized (e.g., "bromite" and "hyposulfite").
As noted in the comments to another answer, in addition to the '-ite' and '-ate' suffixes, there are also the 'hypo-' and 'per-' prefixes, where 'hypo-' only is used with '-ite', and 'per-' is only used with '-ate'. These four combinations are used to span the series of oxo-anions for each element:
- Phosphorus: perphosphate ($\ce{PO5^{3-}}$), phosphate ($\ce{PO4^{3-}}$), phosphite ($\ce{PO3^{3-}}$), hypophosphite ($\ce{H2PO2^{-}}$)
- Sulfur: persulfate ($\ce{SO5^{2-}}$), sulfate ($\ce{SO4^{2-}}$), sulfite ($\ce{SO3^{2-}}$), hyposulfite ("$\ce{SO2^{2-}}$")
- Chlorine: perchlorate ($\ce{ClO4^{-}}$), chlorate ($\ce{ClO3^{-}}$), chlorite ($\ce{ClO2^{-}}$), hypochlorite ($\ce{ClO^{-}}$)
- Bromine: perbromate ($\ce{BrO4^{-}}$), bromate ($\ce{BrO3^{-}}$), bromite ("$\ce{BrO2^{-}}$"), hypobromite ($\ce{BrO^{-}}$)
- Iodine: periodate ($\ce{IO4^{-}}$), iodate ($\ce{IO3^{-}}$), iodite ($\ce{IO2^{-}}$), hypoiodite ($\ce{IO^{-}}$)
For carbon, 'carbonate' was given preference, per the 'most common' rubric, as the only known oxo-anion. For nitrogen, the use of the prefixes was avoided, for what I assume was the sake of simplicity. (I would argue that nitrate and nitrite are both common enough in the natural world that without such a deciding factor there would have been stiff competition for the '-ate' suffix.)
Possible irregularities in structure within a series include peroxo-anions at high numbers of bound oxygens (e.g., I believe persulfate and "perphosphate" are both peroxo species) and variable numbers of bound oxygens for a given oxidation state in anions of heavier elements (e.g., metaperiodate, $\ce{IO4-}$, versus orthoperiodate, $\ce{IO6^{5-}}$, both of which contain heptavalent iodine). As well, there is the possibility for other, non-oxo-anion compounds of the central atoms with oxygen such as chlorine dioxide, $\ce{ClO2}$; nitrogen dioxide, $\ce{NO2}$; sulfur dioxide, $\ce{SO2}$, and trioxide, $\ce{SO3}$; and (of course) carbon dioxide, $\ce{CO2}$.
You can find references to oxo-anions of arsenic, selenium, antimony, and tellurium also (links are to the '-ates'), but to my mind, these tend to edge more toward the behavior of oxo-anion-forming metals, which either only have a single appreciably stable oxo-anion (chromate, molybdate, tungstate, etc.), or break this (hypo-)-ite/(per-)-ate paradigm pretty badly (see, e.g., permanganate, $\ce{MnO4-}$, versus manganate, $\ce{MnO4^{2-}}$).
If you really want to blow your mind, take a look at the polymeric oxo-anions like polyphosphate; or at the at-present theoretical-only orthocarbonate; or at iron, for which apparently all three of the known oxo-anions are referred to as 'ferrate'.
Update, 13 Oct 2023: Following up on Oscar Lanzi's comment, it appears that alkaline earth orthocarbonates have both been predicted and experimentally observed—and, you can apparently buy the tetramethyl orthocarbonate ester commercially$^*$.
$^*$No endorsement intended.
