Hi there, this is a fairly frequently asked question here, so much so that it's in our FAQs. You can check the relevant section out here:
https://www.reddit.com/r/AskHistorians/wiki/language#wiki_alphabetical_order.28s.29
This is not to discourage any further discussion, of course.
Ok, so there are a lot of problems with pretty much EVERY answer both in this thread and in the FAQ on this topic (at least as it pertains to the Northwest Semitic origins of the alphabet). /u/tuviapollack's comment, unfortunately, doesn't cite any sources and misses the mark on a number of issues as it pertains to Semitic origins too.
This is literally my specialty--so I'm going to try and provide a definitive answer with legitimate bibliography (not just Wikipedia articles as in /u/intangible-tangerine's post that appears twice in the FAQ).
Joseph Naveh, The Early History of the Alphabet
Frank Moore Cross, Leaves from an Epigrapher's Notebook
Gordon Hamilton, The Origins of the West Semitic Alphabet in Egyptian Scripts
Albright, W. F., 1966 The Proto-Sinaitic Inscriptions and Their Decipherment. HTS 22; Cambridge: Harvard University Press.
Byrne, Ryan, 2007 “The Refuge of Scribalism in Iron I Palestine.” BASOR 345:1-31.
Carr, David, 2005 Writing on the Tablet of the Heart: Origins of Scripture and Literature. Oxford: New York, NY.
J. C. Darnell, F. W. Dobbs-Allsopp, et al., 2005 Two Early Alphabetic Inscriptions from Wadi el-Hôl: New Evidence for the Origin of the Alphabet from the Western Desert of Egypt. AASOR 59; Boston: American Schools of Oriental Research.
Kuhrt, Amelié, 1997 The Ancient Near East: c. 3000-330 BC. 2 vols. Routledge: New York, NY.
Rollston, Christopher A., 2008 “The Dating of the Early Royal Biblian Phoenician Inscriptions: A Response to Benjamin Sass.” Maarav 15.1:57-93.
---------. 2010 Writing and Literacy in the World of Ancient Israel. Society of Biblical Literature: Atlanta, GA.
Sanders, Seth L. 2009 The Invention of Hebrew. University of Illinois Press: Urbana, IL.
Tappy, Ron E. and P. Kyle McCarter, 2008 Literature Culture and Tenth-Century Canaan: The Tel-Zayit Abecedary in Context. Eisenbrauns: Winona Lake, IN.
Tappy, Ron E., P. K. McCarter, M. Lundberg, and B. Zuckerman, 2006 “An Abecedary of the Mid-Tenth Century from the Judaean Shephelah.” BASOR 344:5-46.
For starters, I've seen the term "abjad" (or, "abgad") used in several places on this sub to refer to the practice of only expressing consonants in the orthography. THIS IS NOT WHAT "abjad" MEANS. Abjad only refers to the ORDER of arrangement of graphemes in a given inscription.
Second, the Ugaritic alphabet was ALPHABET BASED CUNEIFORM. It was INFLUENCED by both Mesopotamian and Hittite writing traditions, but both of the former were SYLLABIC based writing systems--NOT alphabetic. An alphabet is something at or very near a 1:1 correspondence between grapheme (written 'glyph') and phoneme (i.e., the conceptual SOUND represented by the grapheme).
A note on alphabetic origins: the so-called "Proto-Sinaitic" script WAS ABSOLUTELY AN ALPHABET. I don't know why people who comment on this topic in this sub aren't willing to call it alphabetic. It's very clear that (1) yes, it was pictographic in nature and ante-dated the abstraction of graphemes that came later on towards the end of the Late Bronze/Early Iron ages; (2) these pictographs each represented individual phonemes--this is pretty much definitive alphabetic; (3) Garfinkel, et al (in their article on the first ostracon from Khirbet Qeiyafa) buck the notion that this should be called "Proto-Sinaitic" (and I'm inclined to agree with them), I think "proto-Canaanite" is a better and more descriptive term for this early, pictographic alphabet.
These pictographs eventually became abstracted over time and lost many of their pictographic features (e.g., the letter ʿayin was originally depicted with a picture of an eye--the first thing to drop out of this grapheme was the PUPIL, a distinctive pictographic feature). Once these pictographs became more schematized, we start to talk about early linear alphabetic (N.B.: this is still prior to talking about a "Phoenician" script or any of its [i.e., Phoenician's] so-called Tochterschriften).
We don't start talking about Phoenician until the rise of the Royal Byblian kings, the first inscription of which is the bronze spatula inscription of Azarba'l. At this point, no distinct Nationalschriften (e.g., Hebrew, Aramaic, Ammonite, Edomite, etc.) are distinguishable for several centuries and we can only talk about "Phoenician" or "Canaanite" (I personally prefer Canaanite because of the ambiguity of the method of script standardization). /u/intangible-tangerine suggests that the script is called Phoenician "The North Semitic alphabet is more usually referred to simply as 'the Phoenician alphabet' due to the fact that it was the Phoenicians who are credited with introducing the concept to the Ancient Greeks, but 'North Semitic' is a more accurate name as we don't know for sure which North Semitic people used it first." This isn't really accurate. We call it Phoenician because the data is best understood as the Phoenicians having STANDARDIZED the script and caused that standardized form of the script to spread both throughout the Mediterranean and Levant alike. Furthermore, It's NorthWEST Semitic. Not just North Semitic. /u/intangible-tangerine also says that Phoenician and its antecedents were not "fully alphabetic," which is just wrong. There's nothing NON-alphabetic about them in any way. As I said above, pictographic is still alphabetic because each pictograph represents a single phoneme. /u/intangible-tangerine also suggests that alphabetic traditions were incipient in Mesopotamian writing traditions--but this is simply wrong. Mesopotamian writing was syllabic (and logographic) in nature and in no way alphabetic. /u/intangible-tangerine also incorrectly defines "abjad" (corrected above).
Other notes:
-The Hebrew and Phoenician alphabets WERE THE SAME alphabet.
-We're not sure about the original order of the alphabet. There are a few possibilities as attested at Ugarit, Tel Zayit, various acrostic biblical texts (e.g., Psalm 119 and Psalms 9-10).
-The Phoenicians were responsible for the alphabet getting to the Greeks, who then adopted it for their own purposes, turned some consonants into vowels (e.g., /h/ > /e/). Beyond that, I'll leave the history of the alphabet and its evolution into Latin up to Classicists trained in that area of history.
EDIT: a few words