The Diadochi were the Successors to Alexander. They were almost entirely his Companions, his close friends and talented lieutenants. The great general Parmenio and his sons did not survive a possibly untrue attempt at regicide during Alexander's reign, and the only other men to rise (certainly in the earliest days) were Eumenes, Alexander’s secretary (and a Greek, to the contempt of his Macedonian rivals) and Antipater, the elderly and cunning viceroy Alexander had left in command of Macedon.
Alexander had a brother, who was mentally disadvantaged, and a very young son, born after his death. The pair was manipulated by the political machinations of those vying for power and finally slaughtered whilst in Macedon.
The territory of Alexander was enormous. It ranged from modern day Albania, Greece and Macedonia in the west to Egypt and the Arabian Peninsula in the south and all the way to Afghanistan and the Indian subcontinent in the east. When Alexander died the Persian Empire had been comprehensively destroyed, the Macedonian war machine was unmatched, even unrivalled, and he left behind a great number of brave and intelligent generals. Too many, perhaps.
Alexander’s lover and chief of the Companions, Hephaestion, had died before the king himself. Craterus, greatest of the generals, had been sent back to Macedon and thus missed the crucial meeting that decided who ought to succeed Alexander.
In the end a Companion named Perdiccas was named regent, but the mood of the assembly was not convivial and he proved unable to retain the loyalty of the other great men who vied for supremacy. Within a few years the men accustomed to following Alexander and enjoying unrelenting success grew displeased and he was assassinated.
Ptolemy was endowed with Egypt, and proved most resilient of the Diadochi. He lived long, his kingdom (as it became) was never taken from him and he founded a dynasty that lasted until Anthony and Cleopatra.
Antipater had allied with Perdiccas and Craterus, but the latter was surprisingly killed in a first battle with Eumenes, the former was murdered and the decision of Antipater to hand Macedonian vice-regal powers to Polyperchon (a Companion) rather than his son (Cassander) plunged the kingdom into prolonged conflict. Ultimately Cassander proved victorious, and founded Thessalonica (which is named after his wife).
Lysimachus got Thrace (modern day Bulgaria), and Seleucus was fostered in Egypt as Ptolemy’s guest, later to acquire the eastern portion of the empire.
And what of the great heart of the Persian Empire, from Turkey to the Arabian Peninsula? That fell to Antigonus Monopthalmus (‘one-eyed’). He alone had the power to dream of reunifying all of Alexander’s lands under one ruler, and that power caused his rivals to combine their forces against him.
The Battle of Ipsus is not especially well-known, unlike Cannae or Issus. However, it was very important. Cassander did not participate, but did send a significant number of Macedonian heavy infantry (in short supply after prolonged warfare), leaving only a token force to hold his own land. Lysimachus and Seleucus did attend, and the latter brought 400 elephants. Both sides had around 70,000 men, with Antigonus having the advantage in cavalry and his enemies more elephants. Also participating for Antigonus were his son Demetrius and a young adventurer called Pyrrhus, who came from Epirus.
A cavalry charge from Demetrus and Pyrrhus defeated and pursued one of the coalition wings, but they charged too far. The two centres closed and the forces of Antigonus were hard-pressed. Demetrius tried repeatedly to return and attack the enemy centre to help his father, but horses are terrified of elephants and the hundreds Seleucus possessed had formed a screen to prevent the son’s return. In the end, Antigonus and any hope of a unified Macedonian superpower lay dead, and his son and Pyrrhus escaped the field to fight another day.
Seleucus took the lion’s share of the land, and formed the Seleucid Empire (which I think gave way to the Parthians, which were themselves consumed from within by a new Persian Empire).
Cassander retained Macedon, but it never rose to the heights it had once known. Lysimachus and Seleucus came to battle when both were in their 70s, and Seleucus won. He had a hugely powerful empire, and began to march on Macedon. However, he was betrayed and killed by Ptolemy Keraunos (a son out of favour with his father of the same name), who briefly was the most powerful man in the world. He was killed himself shortly thereafter by the Gauls.
It remains to be seen whether Alexander would have been able to keep together his vast conquests had he lived, and whether he would have marched west on Italy. Macedonian politics was soaked with blood (his own father had become king by assuming the throne in the stead of his nephew, but it was Alexander rather than Philip who slew Amyntas IV) and rather unstable.