Hello lelhajj,
This is off-topic and could be another thread, but it is a good question and you deserve an answer. Eastern Catholics can answer this question for you better than I, but I will give you a brief introduction.
The Melkites (or Melchites) of history were originally the supporters of the Byzantine imperial government (and were largely Helenized Greek speaking Christians in the Syria/Palestinian/Egyptian areas) in the first 1000 years of Christian church history. In obedience to the law they adopted the liturgical tradition of the Great Church of Constantinople, Hagia Sophia (the liturgy of St Basil-St John Chrysostom) and worshiped in Greek for many centuries (now in Arabic). In the Syrian region they were generally opposed by the Jacobite Miaphysites and the Maronites, each of which better reflected their own regional ethnicity and language in imperial times as well as a different take on theology.
As Orthodox in the first millenium, the Melkites were the representative Catholic church in the region until the great schism. There is nothing actually non-Catholic in their theology or practices, they are just not Latin.
Today the church is split between the Melkite Catholics and the Antiochian [Melkite] Orthodox. Both churches profess to believe in Orthodox theology but the Melkite Catholic church is in communion with the bishop of Rome and does not seem to formally-officially oppose the unique Latin Catholic theological professions which have developed in more recent centuries.
