Geneticist here.
So first imagine two books, one is Harry Potter the other is Ulysses. The two books are substantially different, right? They are only similar in a way that they are both books and both have very common words like “the” or “he”.
Now imagine two copies of Harry Potter but from different printing. One has a typo that the other doesn’t, so they are slightly different, but the difference is really marginal on the scale of Harry Potter versus Ulysses.
So the differences between species is like differences between two different books (like, HP vs Ulysses). The differences within a species is like typos between the printings of the same book. On that scale, if two people are very different, it’s still the same book but with many typos.
But let’s see how DNA works. First imagine you want to have a necklace with your name. There are those beads with all the letters of the alphabet but you unfortunately do not have those. Instead you only have colorful beads (without letters) in 4 colors. We could come up with a secret code for the alphabet, each letter could be a certain combination of colors. Like, red-red-red equals the letter A, red-blue-red is B, blue-green-yellow is C and so on. Now you can code your name into a necklace using our secret code. Moreover, you could even send me a coded letter using colorful beads.
So DNA works exactly this way, but instead of colorful beads it is a long string of 4 different chemical components usually referred to as bases. The names of the bases are abbreviated with A, T, G and C, and it also uses runs of three to code basically everything in us. So instead of red-green-yellow you would have ATC.
But DNA is also a bit weird because in between the chapters it has long runs of bases where it doesn’t code anything. They are just there for spacing for example long CGCGCGC pieces. In genetics, the parts where DNA has any meaning are called genes, the parts in between genes have no specific name but we will call it spacer for now.
So whenever a woman and a man create a new person, they both copy their DNA and give half of it to the new person. Half mom-DNA, half dad-DNA restores the whole human DNA in the new person. Those half-copies are stored in the egg and the sperm, that’s why those need to unite.
But when the copies are made, mistakes may be made. Mistakes happen usually randomly and there are several forms of them. It can be just a change of a letter, so instead of an A dad-DNA gives away a C. It can be a missing or extra added letter, even a missing or extra long chunk of DNA.
Most of these mistakes happen in the spacer where they make no problem whatsoever. Some happens in the genes, remember, genes are the meaningful message in the DNA. A typo in a gene can be a really big problem like a genetic disease, or just some funny variation. Blue eyes were originally a typo in a gene a couple thousand years ago, before that, everyone had brown eyes.
As I said, copying errors may happen all the time when mom and dad create a new person, but not too many errors. Yet, throughout the history of humans there were a lot of moms and lot of dads so we have a lot of typos in our DNA, accumulated over history. And these typos are basically the differences between you and me.
They can manifest in a form of genetic variances: eye color, blood type, hair type, etc. But although these variations seem to create a huge lot of differences, they are actually rare, compared to the differences in the spacer DNA. You see, spacer DNA has no meaningful message, so any change in there would be invisible. It doesn’t change your eye color or your blood type. Those changes in the spacer DNA are called polymorphism.
So now, how to detect those differences. Some of them, the easiest ones you don’t have to actually detect on the DNA, because you see them. They’re in the genes and you know they are there because of the blood type or similar feature.
The next easy levels are polymorphisms. We invented detection methods before we could really read the whole DNA. You can for example copy a little part of the DNA and check how long it is. Just the length. You remember those long runs of CGCGCG I mentioned? There are other meaningless runs with different lengths. You can copy them out and tell how long is mine, how long is yours. And we have really many of those length differences collected during history of generations. These were the basis of paternity and forensic DNA tests before we could easily read the whole DNA.
And nowadays we have cheap and simple methods to read if not the whole but at least a big chunk of DNA and search for typos in it. There are some DNA services doing it for you, and it’s now really so cheap that many people do it just to figure out their ancestry.
How we can read the DNA is a bit difficult, but it’s never done by microscope or other visualization. Simply because the thread of DNA is so thin, it’s invisible. Reading DNA always involves a machine (there are different machines working on different principles). The common thing is that no matter what machine you use, the result is always just a text file containing A, T, G and C letters. There’s no fancy graphics of rotating DNA like in movies, you just get a bunch of ATGC letters on a computer and try to find differences.
(Note: some simplification was made to keep Eli5.)
Edit: grammar.
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