Some popes had personal collections that they donated to the library. Some acquisitions were the result of favors, such as when the Duke of Bavaria in 1623 gave the Palatine Library of Heidelberg (about 3500 manuscripts) to Pope Gregory XV because of Gregory's political support (they weren't the Duke's anyway: he looted them during the Thirty Year's War).
The Thirty Years War (1618 - 1648) saw a lot of looting of libraries from Prague and Germany, much of which added to the collection of Queen Christina of Sweden. Upon her death in 1689, Pope Alexander VIII purchased her collection, which was essentially the entire Swedish Royal Library. This purchase and re-location turned out to be fortuitous: the Royal Palace where the library had been housed was destroyed in a fire in 1697.
In the Vatican Library, qualified researchers have access to some of the oldest manuscripts in the world, many of them unique.
One of them is the Codex Vaticanus, one of the oldest and most complete copies of the Greek Old and New Testaments. It has been dated to the 4th century. The Gelasian Sacramentary is one of the oldest books on Christian liturgy, with prayers for special masses. The Books of Luke and John from the Lorsch Gospels (Carolingian era) with carved ivory covers reside here (other parts are in the Victoria & Albert Museum and the Batthyaneum Library).
There are also many non-religious texts, such as one of the earliest copies of Vergil, the 5th century Vergilius Romanus, with the Aeneid, the Georgics, and some Eclogues. They also have a 9th century copy of Terence's Comedies, parts of Euclid's Elements, and the poems of Catullus.
The collection also includes interesting selections from outside Europe, such as the sole copy in existence of a 13th century Arabic love story, Hadith Bayāḍ wa Riyāḍ ("The Story of Bayad and Riyad"). The library also contains the pre-Columbian Codex Borgia, one of only a handful of mesoamerican codices not destroyed by Europeans. It is 39 sheets of animal skins portraying information about the Central American gods, rituals, and calendar.
You can also find fragments of a 9th century Old Saxon Genesis, the Libri Carolini ("Charles' books"; four volumes ordered by Charlemagne to refute the Second Council of Nicaea), a 1240s treatise on falconry, a pre-1400 Croatian prayer book, medieval music manuscripts, and over 100 Qurans.
Even if you are not a qualified researcher, you can see some of the collection on their website. In 2012 they collaborated with the Bodleian Library of Oxford to start digitizing a million pages. Since 2012 there have been other projects to digitize more of the collection.
Regarding the Libri Carolini mentioned above: Charlemagne commissioned this in the 790s, had it completed, and then never sent it. What its purpose was and why he decided not to go through with it are worth a look, which we'll do tomorrow.