The first complete alphabet in world history is the Greek alphabet, as it emerged around the start of Ancient Greece, roughly 800 BCE. It is the world's first complete alphabet in the sense that it has a letter to represent each unique vowel and consonant character. The Greek alphabet is not the world's first alphabet in general -- that title falls to the Semitic alphabet, devised by Semitic workers in Egypt around 1800 BCE. Before both the Semitic and Greek alphabets were Egyptian hieroglyphs, which were not considered a complete alphabet, because they had symbols for consonants but not vowels.
Some scholars object to the notion that the Greek alphabet was the first complete alphabet, instead pointing to the Phoenician alphabet, which by convention is taken to originate around 1050 BCE. These scholars argue that others confuse alphabets with transcription systems, and that the Phoenician alphabet in fact includes easy ways of representing vowels.
In any case, the Greek alphabet introduced five vowels: Α (alpha), Ε (epsilon), Ι (iota), Ο (omicron), and Υ (upsilon). You can see that all of these are preserved in most modern languages. English also has the addition of "U." The language also introduced three new consonant letters not present in earlier scripts: Φ (phi), Χ (chi) and Ψ (psi). Three other initially introduced letters eventually fell into disuse: Ϻ (san), Ϝ (wau, later called digamma) and Ϙ (qoppa).
The Greek alphabet as associated with Ancient Greece emerged right after the end of the Greek Dark Ages, derived from the script known as Linear B, which was in turn descended from Linear A, the writing system of the ancient Minoan civilization. The Minoan civilization, which originated in 2700 BCE, was the first major European civilization. The civilization collapsed in 1450 BCE, possibly due to a massive volcanic eruption on the island where they lived, Crete.
The first piece of writing that the Greek alphabet is associated with are the epics of Homer, published around 800 BCE, which are still studied in schools today, as well as commonly being available in libraries. This was the introduction of the modern writing tradition, rather than passing along stories by word of mouth, which by now is approximately 2,800 years old.
Around the time it was initially introduced, the Greek alphabet had several variants. In 403 BC Athens adopted Ionic script for its standard alphabet, and shortly thereafter all competing versions disappeared.