The Library is a prose handbook of Greek myth, and it carries the name of Apollodorus of Athens, a scholar from the second century BCE. But the text mentions a chronicle by Castor of Rhodes, who lived later, so the man whose name is on it can't have written it. The book probably dates from the first or second century CE. Its real author is unknown, which is why scholars call him Pseudo-Apollodorus. It is an ugly name for a very useful book.
What it actually is
There is no poetry in it. It starts with Uranus and Earth and works its way down through the gods, then the great families of the heroes, then Heracles, Theseus, the Trojan War and the returns home. The last stretch survives only as an abridgment, the Epitome, which was pieced together from manuscripts. Everything is told in a flat, clear, practical style. Think of it as a very well-organized set of notes.
Why it matters so much
Many of the old poems are lost, including the Epic Cycle that told the rest of the Trojan story. Where the poets vanished, the Library and a few other summaries are often what we have. It also does something I find endearing: it keeps the disagreements. Again and again you read "some say" and "others say", and it sets the versions side by side without choosing.
It is also the place where you can find things all in one spot. The twelve labors of Heracles are a good example: the Nemean lion, the Hydra, the Ceryneian hind, the Erymanthian boar, the stables of Augeas, the Stymphalian birds, the Cretan bull, the mares of Diomedes, the girdle of Hippolyta, the cattle of Geryon, the apples of the Hesperides and Cerberus. It also tells you something the popular versions leave out. Eurystheus refused to count two of the labors. He said the Hydra didn't count because Heracles had help from his nephew Iolaus, and the stables didn't count because Heracles had asked for payment. So two more were added to the list. I like that the cousin who set the labors was a bad sport.
What it flattens
The price of being a summary is that it has no voice. It tells you what happened but never why it matters. Hesiod's politics are gone. Homer's grief is gone. Euripides' anger is gone. If you only read the Library you could come away thinking myth was a set of facts, who married whom and who killed whom, when really it was a set of arguments.
That is why I say it is a map and not a country. It is superb for finding your way and a poor place to fall in love with a story.
How we use it
This is how we use it on this site. As a first check. If we want to know who Pasiphae's father was, or what order things happened in on the Argo, we look in the Library for the outline, and then we go to the poet for the meaning. It tells you where to look, and that is a bigger service than it sounds.
If you want to read it
Robin Hard's translation, in Oxford World's Classics, comes with helpful notes and is easy to dip into. Read it the way you'd read an encyclopedia: with a question in mind, not from page one.