Yes, there are still Orthodox churches, and they go by "Orthodox Church," although in western countries this may often be preceded by the ethnicity of the founders of the parish - "Greek Orthodox," "Antiochian Orthodox," "Russian Orthodox," etc. A list of the canonical orthodox churches of the world can be found at http://aggreen.net/autocephaly/autoceph.html
Haredi Jews are ultra orthodox Jews found around the world.
The Orthodox Church as it is today was established at the Great Schism of 1054, when Christianity split into the East (Eastern Orthodoxy) and the West (Roman Catholicism).
Orthodox, Conservative, and & Reformed.
The major Eastern Orthodox countries are Greece and Russia. Eastern Orthodox denominations also have significant followings in Serbia, the Middle East and in Egypt, as well as a world-wide diaspora.
If your question is "Why did Jesus Christ found the Orthodox Church", he founded the Church so that we can continue his works here on Earth after his departure.
Russian orthodox. Greek orthodox. Coptic orthodox. Antiochia orthodox. Eastern orthodox. Ethiopian orthodox.
Yes, but the Orthodox look at a different calendar. Christmas Day is on or near January 7 in the Gregorian calendar. For example, Saturday January 7 2012 was Orthodox Christmas Day. More information can be found on the related link below.
The main Orthodox jurisdictions in the US are the Ecumenical Patriarchate, the Orthodox Church of America (OCA), the Moscow Patriarchate, the Antiochian Patriarchate, the Serbian Patriarchate and the Romanian Patriarchate. All of these jurisdictions are in communion with each other. They are not denominations, but simply "sister churches" of the one Eastern Orthodox Church.
Technically there is no such thing as a "Greek Church." Most of the country of Greece is Orthodox. Orthodoxy is a world religion and can can be found in just about every country.
They are called Eastern Orthodox Christians or just Orthodox Christians.
There was no such thing as east and west of Byzantine. The Byzantine Empire was not divided into east and west. Its official religion was Orthodox Christianity, which is sometimes also called Eastern Orthodox to distinguish it from the Oriental Orthodox Churches. The latter is an umbrella term for a number of Churches which hare separate from the mainstream Orthodox Churches found among the Egyptians, the Syrians and the Armenians (the Coptic Church, the Malankara Orthodox Syrian Church, the Syriac Orthodox Church, and the Armenian Apostolic Church).