Here's something amazing: this script originated in Egypt over five thousand years ago. This writing system, this way of recording information so that you and others can benefit from it, is the direct ancestor of the hieroglyphics of ancient Egypt.
It came to you and me by way of the Carolingian empire and any number of typefaces designed since the invention of printing in Europe. And before that, it came from Rome. The Roman script was just one of several in Italy at the time - others included the Faliscan, Oscan, and Umbrian scripts, all used by peoples who were later socii, allies, of Rome. Rome's ascendancy put the otherwise unremarkable script used for writing Latin on the top of the pile.
Where did the Romans get their script from? Well, I've already said that it was Egypt, but there were several intermediaries. The Roman script is just an adaptation of one of the Greek scripts - the Greek alphabets. An alphabet is a script that records both consonants and vowels as separate signs. A, B, C, D, E, and so on. We're used to this system because we grow up using it, but there are actually several different kinds of scripts. Alphabets are very useful for recording languages with words whose meaning depends on changes in vowels, as is true of many Indo-European languages. 'Sing' and 'sang' refer to different points in time simply because of a change in the vowel. Likewise 'run' and 'ran' and so many other verbs. This is true in Greek; Greek words depend on fairly complex combinations of vowels, and failing to record the vowels of a Greek word accurately will cause it to lose meaning.
The first Greek alphabet was one of the first alphabets in history. Not the first script, but one of the first alphabets. It was an innovation: many prior scripts had involved recording consonants only - no vowels, or very few of them - in a system known as an abjad. So the word 'word' might look like wrd, or the word 'thousand' might look like thsnd. This is the way Egyptian hieroglyphs worked in many cases, and it was also the way the Phoenician and several other Near Eastern scripts worked.
This was fine for the languages they recorded. Egyptian, Hebrew, and Aramaic are all Afroasiatic languages, and in most Afroasiatic languages, including Arabic, the meaning of a word is contained in the consonant root. This is a combination of a few consonants with pervasive associations such that if you know the general meaning of the root, you can get the gist of the idea from context (especially if you speak the language fluently and can fill in the gaps). k-t-b in Arabic, for instance, refers to books or studying, so yaktib means 'he writes' and kitaab means 'book'. The vowels are filler (to an extent). So most of the scripts the Greeks came into contact with were abjads because most of the languages depended on consonant roots.
This included the Phoenician script. The Greeks called their own alphabet 'Phoenician letters' in honour of their origins in the Levant. But they changed the script a little: 'aleph, a letter denoting a closing of the throat when your pronounce a vowel (known as a glottal stop), became alpha, a letter denoting the sound 'a. The glottal stop didn't make a lot of difference to meaning in Greek, but it did in Phoenician, and you can imagine a Greek student of the Phoenician script asking how to pronounce the first letter of the alphabet and being told, "'a", and then getting the bright idea of using it as a vowel. Likewise, vowel distinctions didn't make a lot of difference to meaning in Phoenician, but they did in Greek. So the Greeks rather sensibly changed the script to fit their needs, as you'd expect humans to do.
Interestingly, when some Semitic script or other reached India, the locals felt it necessary to do the same kind of thing. Except that instead of taking the abjad and turning it into an alphabet, the Indian scholars turned it into something else entirely: an abugida, where each sign records a combination of consonant and vowel. Ba would be a single sign, as would ghu, pi, and so on, with different diacritics used to indicate vowel quality and length. The basic unit is consonant + 'a' - a, ka, ga, tha, ba, la, and so on. Diacritics were used to change these vowels into ki, gi, thu, bo, lau, and so on. This was an elegant way of recording the vowel differences necessary to make Sanskrit and other Indo-European (and Dravidian) languages in India comprehensible in written form. This was probably an intellectual development of north Indian thinkers who appear to have been used to analysing the phonetics and grammar of the Sanskrit language even before they started writing it down.
Where did the Phoenicians get their script? It seems that they got it from earlier precedents in the Levant. Egypt had had a hand in Syria-Palestine borderlands for centuries before the Phoenicians got involved in regional trade, and had been actively competing for land and power in the area with Mitanni and the Hittites since the fifteenth century BCE. Plenty of Egyptian ideas went north. One of them was the script they used. There were scripts in the area already, most of them written in clay with little wedge-shaped signs ('cuneiform' scripts). Some of these were relatively simple abjads or alphabets, including the script of the Syrian city of Ugarit, which was pretty close to being a pure alphabet. Perhaps these earlier cuneiform scripts, which had considerably simplified the elaborate writing systems of the Near East, served as inspiration for the simplifiers of Egyptian.
The Egyptian style of writing took time to learn, and it wasn't entirely appropriate to the other languages of the Levant. So the locals adapted it to their needs in probably several forms before settling on the 'proto-Sinaitic' script, an abjad capable of recording the sounds used in the Canaanite languages. This was far simpler than the hieroglyphs or hieratic scripts used in Egypt itself. The more-or-less arbitrary choices of a small population of border-dwelling farmers and herders who lived thousands of years ago regarding which symbols to choose from the Egyptian corpus had a significant role in determining the form of the letters you use today to fill in a Captcha or write an erotic novel.
It is from this proto-Sinaitic script that the Phoenician, Hebrew, and Aramaean scripts descend, and it is therefore the source of this script as well. The Phoenician script is little more than a simplification and standardisation of proto-Sinaitic. These scripts were well adapted to the languages of the Near East, but it may be historical accident that enabled certain of these scripts to out-compete the cuneiform scripts rather than an inherent advantage in the scripts themselves.
Where did the Egyptian script come from? Because the Egyptians wrote on papyrus, unlike the Mesopotamians, fewer ancient samples have survived, which means that reconstructing the genesis of the script is much harder to do. Did it have an economic origin? Quite possibly. It has also been suggested that it had a religious origin (the 'hiero-' in 'hieroglyphics'), but I'm not sure about that. An earlier view posited that Egyptian hieroglyphs developed in tandem with Mesopotamian scripts - it is, after all, hard to believe that two different scribal traditions could develop separately in two locations separated by only a few hundred miles of land (this is the view Georges Roux tentatively endorsed, for instance). But given that Egypt was largely separate from Mesopotamia in the early days, with only the presence of Uruk bevel-rimmed pots in northern Egypt attesting to any contacts between the two regions in the fourth millennium BCE, we've got to leave room for the possibility that Egyptian writing, the ancestor of our alphabet, developed on its own as an independent scribal tradition.
The thing about scripts is that they don't develop like language does. Languages develop gradually and it is actually pretty hard to introduce a new word into circulation without any antecedents. This is why etymologists and historical linguists can show the origins of almost all words that exist. They seem to stretch far back into prehistory, with steady changes occurring due to the desire for efficient communication. But scripts aren't like that. You can invent a script and spread it around with little bother. There are clearly precedents to writing systems - it is meaningful to say that this script essentially originates in Egypt - but there is also a high degree of creativity in the invention of scripts. They don't have to develop in a gradual fashion from other scripts. Earlier scripts serve more as inspiration and impetus than as ancestors. We know this is true of some scripts - the Korean script, which is an alphabet, was developed by Korean scholars in the fifteenth century who were not familiar with any other alphabet scripts, but who used the Indian abugidas as inspiration.
This means that the Sumerian cuneiform tradition probably had an impact on the development of this script, in the form of the Ugaritic alphabet. This served as an earlier precedent to and inspiration for the revision of the Egyptian characters, such that proto-Sinaitic developed the way it did due to earlier developments in the cuneiform scripts. Proto-Sinaitic-derived scripts are, therefore, partly the result of the cultural and technological fertility of the eastern Mediterranean in the Bronze Age.
This allows us to look at some traditional problems in a new way. For instance, the Indian scripts are held to be related to our own by way of some Semitic scribal precedent. This is probably true. But scholars have also spent a long time debating how and why they are so different to the Semitic scripts if they're descended from them. This isn't such a difficult problem if it is seen as an invention to solve a problem, with Semitic scripts as the inspiration rather than as an ancestor. Likewise, the Old Javanese script, also known as Kawi, is radically different from its predecessor, Pallava, a South Indian script probably derived from the Kadamba script (and, by way of Brahmi, proto-Sinaitic). It is clear that Kawi isn't just an adapted version of Pallava; it is a new form of script, invented by Javanese scribes for the purposes of recording Old Javanese and Sanskrit literatures in an indigenous manner, and Pallava - and probably several other Indic scripts - probably served as inspiration.
So it's not really accurate to think of writing systems as evolving in the same way as biological organisms, or even languages. They operate differently.
Either way, before the European conquests of the sixteenth century, scripts derived from proto-Sinaitic could be found in Greenland, the Philippines, Tibet, Mongolia, Sulawesi, Assam, Kashmir, Tamil Nadu, the Maghreb, Mali (both Berber and Arabic varieties), Russia, Spain, Scotland, Luxembourg, and almost everywhere in between. Some of proto-Sinaitic's great-great-great-great-great-grandchildren served as precious, sacred scripts in east Asia, primarily for writing Buddhist texts, including Siddham. Egyptian scripts show us the inter-connected nature of Afro-Eurasia in a way few technologies do.
This wasn't a one-way process, either. Egyptian hieroglyphs may have been simplified into proto-Sinaitic and from there into Phoenician and Greek, but these new innovations could be turned on the Egyptian language itself: with Alexandros's invasion of Egypt came Hellenistic culture and the Greek alphabet, which was used, in modified form, to write Egyptian ('Coptic'). The ability to write vowels isn't strictly necessary for many Afroasiatic languages, but it is still quite a useful thing to do, and, in general, the Latin script is better at inscribing Arabic dialects than the Arabic script is (this is part of the reason why the International Phonetic Alphabet is based on the Latin script).
It is an amazing fact that, everyday, you use a technology developed by generation after generation of Afro-Eurasian scholars and merchants in a process that began five thousand years ago in Egypt. This isn't the only ancient technology you use, of course. But it is one of the more obvious and common examples, and I have to say that it amazes me to know this. There's a lot of history in every thing you do, and many things wouldn't be possible without this history. It isn't that you're standing on the shoulders of giants - rather, you're using the collected wisdom of billions of dead humans to achieve your daily ends.