Was it divided into different kingdoms independent of each other, or was it split into regions that each had a different governor? Why did they do it this way? Was his death unexpected?
This is quite a long complex question, but here is a bit of a summary:
Alexander's death at the age of 32, in 323 BC, was unexpected. Although we don't know for sure what caused his death, he only had a short period of 'illness', about seven days.
He left behind one pregnant wife, Rhoxane, but the Macedonians did not know if that child would be born a male, and even if it was, it would be at least 13-14 years before Alexander's son could start playing a role in ruling the empire.
Within the first few days of Alexander's death, the noblemen and cavalry argued with the infantry over who should be the new king, and they eventually compromised by choosing two kings, Alexander's unborn child (who was born a few months later, and named Alexander IV) co-ruling with Philip III Arrhidaeus, who was the half brother of Alexander the Great (this was who the infantry wanted). Philip III Arrhidaeus had some sort of affliction or mental illness (the sources call him 'feeble-minded'), and could not rule independently.
The Macedonians decided that the generals Perdiccas and Leonnatus would be the guardians of the child, while Antipater and Craterus would be responsible for governing Europe.
Most of the governors/satraps in different regions had already been sent to these locations during Alexander's time, and so many of them stayed where they were. There were a few new important appointments, such as Ptolemy being given Egypt to govern.
At first the generals in these different regions were supposed to just govern on behalf of the legitimate kings, but over time they began to act more and more like kings, acquired armies, eliminated their rivals, started founding cities and minting their own coinage, and eventually almost every remaining member of Alexander the Great's family was killed by these men who were supposed to be looking after the empire.
Without any remaining king or members of the Argead dynasty, the satraps/generals were free to assume the diadem/crown, and rule their territories as official kings. Antigonus and his son Demetrius started this in 306 BC, and they were copied by Ptolemy (Egypt), Seleucus (Near East), Lysimachus (Thrace) and Cassander (Macedonia).
There are a lot of ancient sources that cover different parts of these events. Some of the best ancient sources for this time period are Plutarch's Demetrius, Eumenes, and Pyrrhus. Justin, books 13-15, and Diodorus book 18.