Last I heard it was the Canaanites, who developed it on the basis of Egyptian hieroglyphics (or other , simplified but closely related Egyptian writing systems such as demotic). As you probably know, Egyptian writing was a pictographic system, with each word represented by its own symbol. However, Egyptians sometimes had the need to write down foreign words, especially names, for which they had no hieroglyphic symbol. As a workaround they would use a sequence of established hieroglyphic symbols, where the first sounds of the words normally represented by the symbol spelled out (more of less, the foreign word). The Canaanites, who were just round the corner from Egypt in what is now Israel and Palestine, and so in close contact with Egyptian culture, trading with the Egyptians etc. (including, probably, getting enslaved by them sometimes) learned the Egyptian writing system, and adapted this technique to their own, very different language, simplifying the pictographic signs (which had now, in effect, become letter sign) along the way, and thus producing the first real alphabet. The Phoenicians, who lived in the area of what is now Lebanon, just a little further up the coast, learned the idea of alphabetic writing from the Canaanites, and adapted it to their own language, changing the alphabet along the way (as did the Jews, who later took over the Canaanite lands). The Greeks then got the idea of alphabetic writing from the Phoenicians (with whom they had trade relations), adapting it to their own language again, particularly by adding new signs for vowels, which had not been represented before.
Presumably the Phoenicians tended to get the credit for the original invention of the alphabet in the past, because the Greeks took the idea from them (with acknowledgment), and the Western world, for a long time, owed most of what it new about ancient history from classical Greek authors (who may not even have have known where the Phoenicians got eh idea from). Furthermore, Canaanite culture was more or less extirpated long ago, and the Old Testament painted them as bad guys, because the Jews had to conquer their lands to establish themselves. But more modern research has reveled them as the real inventors of the alphabet, and the Phoenicians moat important role was in transmitting the idea of an alphabet (and many, but not all, the basic letter forms) to the Greeks. From whence the idea passed to the Romans, and eventually to us.