The decline of the Byzantine Empire was a long and complicated process that is usually said to start with the battle of Manzikert and its aftermath which I talked about in length here: http://www.reddit.com/r/AskHistorians/comments/2wnwbz/if_the_battle_of_manzikert_was_not_a_bloodbath_as/cosm847

The short summary is that the civil war that resulted from that battle allowed the Turks to start settling in Anatolia permanently, but powerful Byzantine Emperors like Alexis, John, and Manuel Komnenos kept the Turks in check.

The second event generally associated with the decline of the Byzantine Empire was the Fourth Crusade. The idea of the Fourth Crusade was to have the city state of Venice provide a fleet to carry the Crusade to Egypt, which was the center of Muslim power in the Near East. However, fewer crusaders arrived than anticipated, leaving the Venetians in a financial crunch.

Meanwhile: The Byzantine Emperor Alexis II Angelos had recently been overthrown by his brother Alexis III. Alexis II's son Alexis IV escaped Constantinople and eventually meet up with the crusade, offering financial support if the Crusaders would help to put him on the Imperial throne.

Long story short (Roger Crowley's City of Fortune provides a good overview of the Fourth Crusade and the Venetian Colonial that spring out of it): the crusaders win the throne for Alexis IV, but it proves impossible for the emperor to prove the promised finances and becomes unpopular in Constantinople due to the finical drain. Eventually, Alexis IV is murdered by his uncle Alexis V Doukas who becomes emperor and refuses to give the crusaders any more tribute at which point they attack and loot the city.

In the aftermath of the sack of Constantinople, a Latin Empire is set up with Baldwin of Flanders as Emperor. The formerly united empire then frag-mates into crusader kingdoms reminiscent of those in the Levant, Venetian and later Genoan colonies like on Crete, and three Byzantine successor kingdoms: Epirus, Nicea, and Trebizpod.

Eventually, the Empire of Nicea does reclaim Constantinople with Genoan help (Venetian-Genoan rivalry being a huge thing in the 13th and 14th century Mediterranean). However, this reborn Byzantine Empire is never able to reclaim the holding lost in the Fourth Crusade and some, like Trebizond, would outlast Byzantium itself (though by a few years).

Having gone though the outline of the Fourth Crusade and its outcome, I would say the removal of Alexis II by Alexis III and Alexis IV's appeal to the Fourth Crusade are what led the empire into a spiral of decline from which it never recovered.

Manzikert can be seen as the beginning, especially since it led to the Levant crusades in the first place, but Byzantium had experienced crises just as threatening in the past ( the Persian war of 602-628) and by and large, the empire had reached a point of stability before this point.