Not all temperament tables provide the same information, nor do they all provide it in exactly the same way, but the following properties should cover most needs.
The subgroup (or domain basis) of a regular temperament is the set of all intervals which are considered to be approximated by the temperament. For example, it is common to consider that 3/2 is approximated by 12-tone equal temperament, therefore 3/2 would be included in this set, but other intervals like 11/8 could be excluded. Most of the time, a subgroup exclusively contains just intonation (JI), aka rational, intervals.
In a subgroup, all intervals must be reachable by stacking (up and down) copies of a few "generating intervals", or generators. Continuing the previous example, if 3/2 is taken as a generator of the subgroup, then 9/4 is also included in the subgroup. If 2/1 is added to the list of subgroup generators, then intervals like 4/3 can be reached by combining a 3/2 down with a 2/1 up (i.e. 2/3 × 2/1 = 4/3).
The generators of the entirety of JI are the infinite set of prime numbers: 2, 3, 5, 7, etc.; therefore the most common type of subgroup of JI uses a subset of primes (or, if 2 is in the subset, equivalently octave-reduced prime harmonics such as 3/2, 5/4, 7/4, etc.) as its generators. A subgroup is generally expressed as a list of its generators separated by dots: e.g. 2.3.5 is the subgroup of all intervals consisting of combinations of 2/1, 3/1 and 5/1. The 2.3.5 subgroup is equivalent to the 5-limit, the subgroup defined by the prime harmonics up to 5, though for maximum clarity the temperament tables currently prefer spelling out the primes explicitly.
However, it may be reasonable in some cases to include composite numbers in a subgroup: the subgroup 2.7.9.11.15 includes some intervals that contain 3 and 5 in their factorization (such as 9/7, 15/8, or 5/3 - the last being interpreted as 15/9), but not others (it would not contain an interval like 3/2 or 5/4, since these can't be reached from multiplying and dividing 9 and 15 with primes); or even fractions, like the subgroup 2.3.11.13/5.17 (note that this is interpreted as 2.3.11.(13/5).17), which includes intervals of 13 and intervals of 5, but only when a power of 13 is matched by an equal power of 5 on the other side of the fraction. Composites or fractions treated as primes in this context are often called "formal primes" or "basis elements."
An abstract regular temperament can be thought of as a family of valuations (tunings of the temperament) of the primes in its subgroup that satisfy certain equations; if we bold the numbers to make it clear that we are speaking of them as abstract variables, an example of such an equation would be 34 = 24 × 5. Each of these equations corresponds to setting a JI interval to be equal to the unison (1/1); the equation specified here sets 81/80 (with factorization 2-4 × 34 × 5-1) to the unison, in other words tempering out 81/80.
The comma basis is a list of such intervals - commas - that are tempered out by the temperament, thereby restricting the set of possible tunings. If a tuning only tempers out one comma, then the only intervals within the subgroup that are set to the unison are that comma, and its positive and negative powers (in the case above, 80/81, 6561/6400 = (81/80)2, etc.); therefore there is only one logical choice for "which comma" you claim is tempered, that being the simplest of these powers greater than the unison. Rank-2 temperaments in subgroups including 3 primes, such as the 5-limit, only temper out one comma, for instance.
However, if a tuning tempers out multiple independent commas, the situation gets more complicated, for the set of tempered intervals in fact forms a lattice generated by more than one generator (in other words, a nontrivial subgroup of JI), and the choice of which specific intervals to consider generators (which in this context are basis commas) is not always obvious. For instance, septimal meantone tempers out the intervals 126/125 = 2 × 32 × 5-3 × 7; 225/224 = 2-5 × 32 × 52 × 7-1, and 81/80, but 81/80 = (126/125) × (225/224), and therefore these three commas are not all independent - but all of them are useful, in that all three define prominent temperament families (collections of regular temperaments that share a tempered comma in common): 81/80 defines meantone, 126/125 defines starling, and 225/224 defines marvel. Various methods exist for choosing which commas are selected to be basis commas, which are associated with the technique of matrix echelon forms; in the case of septimal meantone, the basis commas are chosen to be 81/80 and 126/125 at the price of obscuring the fact that it also tempers out 225/224.
As a last note, factorizations are generally abbreviated in the form of a (subgroup) monzo, which is simply a list of the exponents in a factorization that are attached to each (formal) prime in the subgroup, so that for instance 225/224 would be [-5 2 2 -1⟩ (in this case the subgroup is 2.3.5.7; it should be specified if there is any ambiguity, but if not it can be assumed to be the temperament's subgroup).
