Edit: In EU4, A Personal Union is where two (or more) states are ruled by the same monarch while staying fundamentally different countries with their own laws, government, etc. and given enough time and effort one state could be fully integrated by the other.
In the game, a Personal Union is represented as one state becoming the overlord of another due to the monarch of the first state claiming the throne of the second, usually through inheritance. The second state acts much like a "vassal", or "junior partner" of the first, unable to fully dictate foreign policy and dependent on the first state for progress. After enough in-game time has passed, the first state can try to integrate and annex the second state into the first completely.
There were clear examples of the personal union phenomenon (one King claiming to rule several separate, autonomous realms) in the Muslim world, but they had largely ended by the time of EU IV, and the later examples (like the late Mughals) were fleeting.
During the time of the Crusades, the ancient power of the Caliphate had been broken to the point at which the Caliph was just a rump ruler in the city state of Baghdad. He still, however, commanded immense moral authority and could legitimize rulers in the Sunni world. Nur ad-Din Zengi, a Turkic warlord ruling Damascus, was the most prominent founder of a Muslim personal union, ruling between the First and Second crusades. He inherited territories in the style of Charles V from dead relatives, quickly becoming Atabeg not just of Damascus, but of Mosul and much of the Levant as well. Later in his life, he invaded Egypt and proclaimed himself Sultan - he retained many local institutions, including the office of Vizier.
However, he never claimed the title of Sultan of a united Zengid Empire (the word is largely a modern invention), nor attempted to unify his lands into one bureaucracy or system.
Saladin, who revolted against the heirs of Nur ad-Din, proceeded similarly, conquering old Zengid territories (which were split now between Nur ad-Din's heirs), but proclaimed himself Atabeg or Sultan respectively of the separate territories, instead of creating a unified government.
Both rulers petitioned the Caliph constantly for appointment as Sultan or Atabeg. These titles are important. Sultan literally means executor of the orders, or 'authority', or in some cases 'deputy', and an Atabeg is a lord. Both were theoretically subordinate to the collapsed power of the Caliph.
A similar situation repeated during the withering of the Mughal Empire in India. Court plots and corruption had rotted the Mughals after the death of Aurangzeb, and Asaf Jah I became Viceroy of Deccan, before also receiving an appointment as Nawab of Assam - both titles legitimized by the Mughal court under his petition. Asaf Jah followed the pattern of Saladin and Nur ad-Din, being a single ruler over separate governments, but never founding his own centralized, unified "Deccani Empire".
As we can see from these examples, personal union arises when an area shares a source of legitimacy. In Europe, this was bloodline, and for a while, Papal permission. In the medieval Middle East, this was the Caliph's blessing. In the declining Mughal Empire, this was appointment by the Peacock Throne, however weak and degraded it may have become. Personal union is not as simple a business as "you are next in line, so you are King of us all"- in fact, it's very complicated. A good example is the unification of Castile and Aragon, two nations with vastly different institutions. A huge amount of scheming and politicking was necessary to secure a unified succession, and even so, revolts were common. However, the rise of one King as monarch of both countries with their different governments was possible because of the shared belief in legitimacy through bloodline. Personal union can be seen as a declaration that "we are two different governments, but we agree to both pledge to you, because you are legitimate, but we will not agree to merge as one government".
Going off of this, an era of personal union ends when that source of legitimacy no longer becomes relevant - through catastrophe, new sources of legitimacy. After the sack of Baghdad by Hulagu Khan and the Mongol invasions of the Middle East, the extinction of the Caliphate and mass devastation led to the foundation of new states with new sources of prestige and legitimacy. The Mamluks claimed the Caliphate for themselves, the Safavid Persians derived their legitimacy from being the defenders of Shi'a Islam, and the Ottomans derived their authority first from being a mix of holy warrior Kings and successful raiders, and next from appropriating the institutions of the Byzantines. It would have been inconceivable that an Ottoman Sultan could proclaim himself "Shahanshah of Persia" and received any support whatsoever from the Persian nobles, even if he had an impeccable claim by bloodline: the two nations' sources of legitimacy were too different.
In Europe, this situation took a longer to unfold. The "right of blood" wasn't discredited as an institution or a source of legitimacy, and no continental catastrophe forced new charismatic leaders to rise and create their own sources of legitimacy, until the gradual dismantling of hereditary rule in the 19th century.
(Sources:
Saladin, Nicolle
Saladin, John Man
Advanced study of the history of modern India, Mehta
Imperial Spain, JH Elliot)
If you could elaborate on a PU, what it does in the game, when it becomes available by research, and how it's used that might be helpful.