What it actually was
The Library of Alexandria was not one library but at least two institutions: the main library attached to the Mouseion (“shrine of the Muses”) in the royal Brucheion district, and a smaller daughter library attached to the Serapeum temple on a hill in the western part of the city. Both were part of a research-and-education complex funded by the Ptolemaic dynasty, which had ruled Egypt since the death of Alexander the Great in 323 BC.
The Mouseion employed approximately a hundred resident scholars on royal stipend. Among the most famous were the geographer Eratosthenes (who served as chief librarian around 245 BC and measured the Earth’s circumference), the mathematician Euclid (author of the Elements), the astronomer Aristarchus, the philologist Aristophanes of Byzantium, and the poet Callimachus.
How many books
Ancient sources give wildly varying numbers for the collection — from 200,000 to 700,000 scrolls. Modern estimates, based on the physical capacity of buildings the surviving sources describe, place the peak at approximately 100,000 scrolls. A scroll typically contained the equivalent of one chapter or one book of a modern volume; the Iliad, for example, would have filled roughly 24 scrolls. Translated to modern reading material, the Library at peak probably held the equivalent of 30,000–50,000 modern books.
How it actually ended
The myth that the Library was destroyed in a single fire is wrong on multiple counts. There were several distinct events, and the loss was gradual across six centuries.
48 BC — Caesar’s fire. Julius Caesar set fire to the Egyptian fleet during his war against Ptolemy XIII. The fire spread to dockside warehouses and probably destroyed scrolls being stored there for export. The main Library in the Brucheion was untouched — Strabo describes it as still functioning in 25 BC.
273 AD — Aurelian’s destruction of the Brucheion. During the Roman emperor Aurelian’s reconquest of Alexandria from the rebel queen Zenobia, the entire Brucheion royal quarter was destroyed. The main Library almost certainly perished in this event. The Mouseion is never mentioned again as a working institution after Aurelian.
391 AD — Destruction of the Serapeum. The Christian patriarch Theophilus organised a mob attack on the great pagan temple of Serapis, demolishing the structure. Whether the daughter library was still in the building, or had been dispersed, is contested.
642 AD — the apocryphal Caliph Omar story. The widely-repeated story that the Arab conqueror Caliph Omar ordered the Library burned after the 642 AD capture of Alexandria appears in no contemporary Arab or Christian source. It first surfaces in the 13th century, six centuries after the supposed event. By 642 there was no library left to burn.
What was lost, and what wasn’t
Most of the actual scholarship done at the Library — the editing of Homer, the geographical measurements, the mathematical proofs, the astronomical catalogues — survived because the work had been copied and disseminated across the Mediterranean while the Library still existed. Euclid’s Elements, Eratosthenes’ geography, Ptolemy’s astronomy all reached the modern world via medieval Arab and Byzantine manuscript traditions.
The actual scrolls of the Library are gone. There is no surviving fragment that can be identified as definitely from the Mouseion. A single inscribed limestone block possibly from the east colonnade, recovered in excavations beneath the modern Bibliotheca Alexandrina in 1995, is the only known physical trace.