Mage Rank and Linker Cores
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Ranks run from F to A to SSS in terms of magical ability, with lower than F- being no magical ability at all. It should be noted that Ranks are measures of practical ability not solely brute power or technical skill; they are assessed exams; you are a B-Rank because you pass the B-Rank exam, they are not intended as measures of your Linker Core.
Genetics plays less into this than you'd think. The baseline in most of Dimensional Space is capable of reaching A-rank or greater. It just requires a gruelling training regime from a fairly young age, constantly pushing them to the limits of what they can do as to promote the development of their linker core, just like building up muscles, and, simply, most people don't want to be soldiers. Genetics are not the only thing that counts in most of dimensional space, and that's a damn good thing, because if it was, the TSAB would have a few field officers for each planet.
And then, yes, you get people like the known 97ers, who are the metaphorical people born with the ability to bend iron bars in half. Which has its own issues; some people in the TSAB have speculated whether the median level of magical ability would be higher if you didn't have those genetic freaks, who breeze through things that other people have to slog through for years, and serving to dissuade other people from trying their best because "they don't have the genetics", so they'll never be able to be as good.
That's one of the reasons for the Ground Force's dislike of the Air Force, actually. The Air Force tends to recruit younger, from more naturally powerful mages, and so it's sort of a bastion of "genetics is the key factor" thinking. The Ground Force is rather more "effort-based", because they have to be, and so tend to find the Air Force to be rather patronising and filled with young idiots who don't even need to shave properly yet in field commands.
And that's been found to a standing problem in some TSAB military operations. Young, naturally talented mages do, despite the best efforts of the system, tend to be overpromoted relative to their actual experience and accelerated into officer training, and problems do arise when you have a 17 year old AA-rank leading 30-year old A-ranks (and when a 50 year old is being out-ranked by a 19 year old Colonel). True or not, there's certainly an edge of resentment for how powerful mages get treated, and there have been a few military disasters where overpromoted mages have misjudged situations through lack of experience, and either had to blast their way out, or been unable to blast their way out.
And Teana is a canon example of how magical power isn't everything, and how playing smart allows you to punch quite a bit above your weight class.
Broadly speaking, though, S rank and above means that you've changed the combat paradigm in some way that means nobody less than a rank under you can even fight back on any sort of realistic basis. AAA is the highest you can get while fighting "like other mages, but better" - S rankers have generally developed their own microstyle within the style they use which is full of fairly unique tricks. Zest's insane speed/power bursts are an example of that, as is (once she gets that far) Fate's high-speed Mid-style-at-close-range, which is a horrible surprise to Mid-stylers while still letting her use ranged attacks better than most Belkan users. Very tricky to pull off, though, and comes with some big downsides, one of the key ones being defence.
Note that rank isn't just one thing. Megane - and even Nanoha - can pull off specific S rank spells. Fate's speed would probably be classed as S rank, since not even Zest can match it, and Hayate has a bunch of them. Zest can probably outclass Precia in a straight-up, close-range fight (made easier by the fact that his rank is all about murdering people in the face, whereas hers is across a wide range of fields of which combat is only one, and not really an important one). But being acknowledged as an S (or SS) rank mage is more than just being able to cast magic at that level, it's being at that level overall. Hayate can't even generate her own Barrier Jacket without having Reinforce unisoned, so soon after rebooting the Book. She may be able to pour on the firepower in S rank quantities, but she's not well-rounded enough to qualify for the rank itself. Way too many holes and weaknesses in her skillset.
At this level of power, the definitions get a bit fuzzy. S+ rankers are unique, incredibly rare and usually defined by how they break the paradigm in some fashion. You can't really put anything more than the vaguest of general trends around them, and the rank system itself is intentionally hazy on pinning things down exactly, to account for how different people have different talents and how mages aren't at exactly the same level in all the fields they know.
You could say that to earn rank X, you need to have most of the major areas relevant to the type of rank you want covered at X rank level, and no areas below about X-1 level. You can see how this leads a lot of mages to specialise somewhat, with A rank shooting skills in a C rank sniper who isn't very good in any other formats, and how a high-rank mage is scary for that, because they don't really have any massive weaknesses in an area you can exploit; they're tough all round.
- In practise, only one person has ever achieved that standing, and is in fact whom the SSS rank is measured against.
The Linker Core is an intangible organ nestled a little below the heart and between the lungs, near the solar plexus. To call it an organ is not quite accurate terminology - the Linker Core is the point at which mana generated by the whole body coalesces into a usable form, creating a nexus, detectable as a "dimple" above and below the body in Dimensional Space, from which it flows out back into the rest of the body. Damage to the Linker Core is dangerous, and can seriously affect the cycle of mana through the body and adversely impact the ability to use it in magical spells.
Linker Cores and Mage Rank are not completely linked; for instance, even on Earth (with an unusually high number of non-magical citizens), it would be rare to find someone without a Linker Core any more than it would be likely to find someone naturally missing any other organ.
During pregnancy, the mother's linker core 'buds' and forms the child's linker core. Consequentially, many Rare Skills can only be inherited through matrilineal descent.
Linker Cores cannot be found in wildlife, though creatures from Type-3 worlds have structures analogous to them. And the reason they have those is that there's a larger amount of ambient mana there than on any any other world-template; they evolved to take advantage of a naturally occurring resource, basically. No Alhazredian messing around required.
People with affinities basically find it easier to manipulate mana in certain ways - the archetypical example being Fate's lightning affinity, which makes it really easy for her to turn mana into electricity and vice versa. Signum's fire affinity makes her frighteningly good at turning doing the same thing with heat. There are "ice" affinities, which are more like a half-working heat affinity where they're really good at converting heat to mana, but can't really do the same in the opposite direction.
To a less noticeable extent, there are also affinities for other things - there are people who are astoundingly good at converting between mana and kinetic energy, like how Nanoha has a knack for spells that move stuff or transmit force. That's not affinity-level, though, that's just being relatively good at them. There aren't affinities for things like "earth" or "water", and you can't create matter from nothing with an affinity - it would be more accurate to say that an affinity is a natural talent at converting between mana and a different form of energy.
Affinities are, surprisingly, not an engineered trait. Well, some of them are, but the phenomenon has been naturally occurring for as long as humans have had magic, and is indeed one of the earliest and most primitive forms of magic that humans discovered. Thousands upon thousands of years ago, before spells were even conceptualised, the advantage was held by the tribes with "spirit-touched" shamans who could conjure lightning from their hands, blow a man twenty feet away with a touch or walk unharmed through a blazing fire. Those primitives were the ones who first began to learn how they did so, and draw on the powers that so naturally bent to those purposes, and instead use them to crudely enact their will upon the world, making themselves stronger, faster, wrapping themselves in protective fields - and crucially, those first founders of what would become Alhazred were the ones who taught others of their tribe how to do so, giving rise to the first real mages.
And the rest, as they say, is history.
And yes, that does mean that there are also sound and light affinities. In fact, the latter, there's a common joke when a baby is screaming loudly of going "sonic affinity?" Their "output" forms aren't approved of nowadays because they're indiscriminate, deafening or blinding everyone in the vicinity regardless (though Barrier Jackets countermeasures work against that). In ancient times, though, someone with a light affinity was gold, because they could blind half the charging enemy forces in one massive flash. Or sound, to deafen everyone nearby. Nowadays, though, it's usually the passive form - there'll just be a hole in the sound or light temporarily as they cast. It doesn't give much, but it's a trickle of mana towards every spell, and every little helps.