It says a lot about the state of discourse and gender science that I can't tell if you genuinely don't understand or if you are being facetious.
But assuming you are being genuine: your sex (faka gender) is encoded in your DNA on the 23rd chromosomal pair. If you have two X chromosomes (XX), you are a female human (woman). If you have one X and one Y (XY), you are male human (man). These chromosomes dictate your sexual development (organs/hormones/etc). There are some people (estimated at 1-2% of the human population) who are "intersex", in that they don't have an XX or XY pair, but some variation thereof (XXY, XYY, etc). However even these variations generally fall under one or the other umbrella of male or female. e.g. Klinefelter syndrome (XXY) is generally going to present as male (albeit frequently with poorly developed testicles and other complications). Genes are complicated so this is a bit of an oversimplification.
To make this more about sex itself (since humans are a sexually reproducing species and that is why we are having this conversation at all), there are effectively three options for all humans: you produce small gametes (sperm / male) or you produce large gametes (eggs / female), or you produce neither due to some complication and therefore cannot participate in procreation. This is dictated almost wholly by your DNA.
So, back to the article: they know the DNA in the deer tooth was from a woman because (presumably) the 23rd chromosome was an XX pair.
Just to add one more wrinkle to the link between DNA and the presentation of sex, there is also androgen insensitivity syndrome, in which the person has a Y chromosome, but the developing body does not react to the testosterone produced by the testes.
The result is someone who may appear to be female, have female genitals, but has testes where ovaries would be expected. It's often not discovered until puberty doesn't happen as expected, and they discover they can't have children.
Like all these conditions, this is on something of a spectrum depending on how total it is.
As you say, genes are complicated. So much more so than is often implied in reporting of discoveries of "a gene for X".
I was a little blithe with my reply here, and feel bad about the scientific inaccuracy I'm perpetuating. The "everyone in between" spectrum often has strong signals, but sometimes it's not very strong straight from DNA alone! And it's also incredibly complex, far far more complex than doing mutation calls, to the point that automating it is quite difficult. Sure, finding XX, XY, XXY, and the many other combinations of sex chromosomes as they appear in nature is easy. Finding subtle inactivations of the genes on the Y chromosome is much harder. In fact analysis of genomes of intersex people is a common use of consumer whole genome and whole exome sequencing, so any consumer sequencing company that simply reports XX as female and XY as male is laughably amateurish, and your customer support lines will be swamped with extremely knowledgeable self-educated intersex people that will being providing tons of information for the company to up their game.
> so any consumer sequencing company that simply reports XX as female and XY as male is laughably amateurish
To my knowledge there is no XX person who can produce sperm and there is no XY person who can produce eggs. So while "XX is female" is over-simplified, "XX is not male" seems fairly clear.
> your customer support lines will be swamped with extremely knowledgeable self-educated
In no other field this combination (self-educated, actually knowledgeable, and calling customer support) seems to exist, so that alone might be worth further study.
They are highly motivated people trying to understand the genetic basis of themselves better, and have usually been seeing doctors throughout their lives towards that end, and usually they'll know more than their doctors because of the rarity of their phenotypes. I have tons of respect for that.
