The Latin Empire represented the attempt of the Crusading Western European states to supplant the Byzantine Empire, the last remnant of the classical Roman Empire, centred on the city of Constantinople.
Notionally Byzantium, although Eastern Orthodox in its Christianity, supported the Western, Catholic, drive to recover the Holy Places in Judea from the Muslim armies that had seized them. In practice, the Byzantines found the Muslims as honourable as the Catholics (if not more so), and far better as trading partners (due to them being closer and having more ready cash to offer than the Europeans, who generally, while wanting the goods of the East, wanted to keep their money closer to home).
So the Eastern Emperors had slowly moved closer to the Muslim side. This irritated the West, whose perception was that the Emperors had sold out to the Saracens, and were both profiting from the Crusades by selling food to Crusading armies at extortionate prices, and also by then selling details of their routes of travel and of their dispositions to the Muslim lords, to assist in their ambush (whereupon, so it was thought, Muslim and Byzantine shared the loot).
So, in 1204, the soldiers of the Fourth Crusade attacked, and sacked Constantinople, expelling the Imperial House and killing Emperor Alexius, and set up their own empire, as a Catholic state. The first Emperor was Baldwin, hitherto Baldwin IX Count of Flanders. Boniface of Montferrat, one of the Crusade's leaders, had hoped to be elected, but he was seen as too closely linked with Alexius.
The Imperial House, meanwhile, went on to establish new centres of power, with empires in Nicea and Trebizond and a despotate at Epirus, but the Latins were able to take power across much of Greece and north-west Turkey.
Baldwin lasted less than a year, being killed in battle, and was succeeded by Henry of Flanders, his younger brother. Henry ruled for 11 years, successfully defending his borders against both the Niceans and a Bulgarian Tsar, and his authority against the Catholic clergy. He died in 1216, possibly of poisoning, perhaps at the hands of his Greek wife.
Leaving no sons, Henry left a troubled succession. His brother-in-law, Peter of Courtenay, had Crusaded, and was selected as Emperor. He obtained consecration at Rome and set out. His route, however, took him through Epirus, where he was captured, imprisoned and died. His wife, sister to Baldwin and Henry, Yolanda by name, did reach Constantinople, and ruled as regent in her husband's stead and, after his death, briefly in her own right until she died in 1219. Her elder son, Philip, declined the Imperial throne, which therefore fell to her second son, Robert of Courtenay, then in France. It took him until 1221 to reach Constantinople, but in the interim the Nicean emperor had seized lands, despite the best efforts of Conon of Bethune as Imperial regent. Robert, as Emperor, proved unequal to repelling the Niceans, and was compelled to recognise John Ducas Vatatzes as their Emperor, and his realm as sovereign. Robert then threw himself into a matrimonial entanglement, marrying the fiancee of a Burgundian who, taking offence, drove Robert out.
Robert left no heir, but Yolanda had had other sons, and the 11-year-old Baldwin was selected as successor Emperor (the Burgundian being unable to parlay matrimonial disappointment into an Imperial title). While he grew up, the titular king of Jerusalem (the holy city being by now firmly in Moslem hands), John of Brienne, served as regent, and he governed until Baldwin II was twenty years old. Hopes that John, perhaps, could throw back the Niceans was fruitless, and by the time that Baldwin was installed as Emperor his Empire was little more than the city itself. In 1234 he married John's daughter Marie, and in 1236 he went to Europe to round up support for his troubled Empire. John, meanwhile, had died, and although Baldwin II returned with an army, he could do little with it.
The rest of his life was in like pattern, touring Europe desperately seeking military support, and being rebuffed. He even mortgaged his son, Philip, to Venetian merchants, for money, but to no avail. In 1261 Michael Palaeologus, the Nicean co-Emperor, seized Constantinope from its absentee Emperor (and celebrated by having his co-Emperor blinded and banished, leading to his own excommunication by the Orthodox patriarch). Baldwin made his way to France, then to Italy where he and his son (redeemed by Alfonso of Castile) lived on pensions from Charles of Anjou. In 1273 Philip married Charles' daughter; a few days later Baldwin died.
Between 1272 and 1383 the title of Latin Emperor was passed on, through Philip's daughter to Charles of Valois, through his daughter to Philip d'Anjou, and then down his line, until it reached Louis of Anjou in 1383, who never troubled to use the title.
Representing a brief burst of Western effort against their fellow Christians, all that the Latin Empire really achieved was to afford the Byzantine Emperors a chance to re-learn the skills of living "in the wilderness", and to fragment their Empire and make the eventual Moslem conquest a fraction easier.