Many emperors had founded or re-founded cities, and the empire in the third and fourth centuries had several simultaneous "capitals". But Constantine actually moved the whole government and bureaucracy from Italy to Constantinople, and the new city soon became the primary capital of the empire and the real heart of Roman and, rather quickly, of "Byzantine" civilization. It soon outstripped even Rome in beauty and prosperity, and by the fifth century its fortune had become identified with that of the empire as a whole.
Constantine himself apparently laid out the general plan of "Nea Roma" -- New Rome -- and, according to some ancient sources, he actually ploughed the line of the new pomerium, duplicating the rituals that Romulus had performed on the Palatine 1082 years earlier. He set a five year deadline for construction, and, on 11 May, 330, only a little beyond Constantine's five year planning date, a great ceremony was held to dedicate Constantinople. In general, the city was patterned after Rome. Before long Constantine's new city, rather than Rome, became known simply as "The City."
Constantinople was located on a peninsula on the European side of the Bosphoros (the western side of modern Istanbul, Turkey). It was ideally situated between the eastern and northern military frontiers. It was already an important center of trade, located on major north-south and east-west trade routes.
The Golden Horn was on the north, the Sea of Marmora on the South, and the Bosphoros to the east. Walls protected the city from attack along the west; sea walls were incorporated only later. Wide avenues ran from the gates in the wall eastward toward the center of the city; many of these streets were colonnaded and lined with impressive public monuments; the most splendid of these was the Mess (from "mezzo"), or the "Middle Street." There were seven churches, and an imperial residence, which later became the central part of the Great Palace. The palace complex lay near the site of the ancient acropolis at the Far East end of the city.
By the middle of the fourth century Constantinople had become the legal equal of Rome and it had most of the same institutions, including a Senate. The importance of the city was enhanced by the emperors from 395 to 611, few of whom left the city for extended periods. Monuments of classical art were brought to the city to decorate it. The construction of the Theodosian Walls in 413 was an important milestone in the development of the city; they enlarged the area of the city and enclosed it in triple, almost impregnable fortifications.
And what of Rome after the Imperial departure? Constantine is only known to have returned to Rome twice, to mark the tenth and twentieth anniversaries of his conquest. The population declined rapidly: anyone whose livelihood or prestige depended on the emperor, the government, or the bureaucracy followed Constantine to the east. The number of Romans eventually bottomed at about ten thousand -- down from one and a half million at Rome's ancient peak -- but that was almost a thousand years later, after barbarians and plagues also had taken their toll. Metropolitan Rome shrunk down into the Campus Martius on the Tiber's east bank.
There were still emperors in the "Western Empire" -- although perhaps in Milan or Ravenna rather than Rome -- until 476 when young Romulus Augustulus, the son of Attila's former chief-of-staff Orestes, was sent packing after less than a year on the throne. He and his mother started an abbey near Naples. Rome's new ruler, Odovacar (Odoacer in Latinized accounts -- long assumed to be a Goth, but called a "Skyrian" in contemporary writings), wanted the city, but not the Imperial title. Fortified towers owned by powerful "princely" families (parts of which are some still standing) surrounded by "borgos" became the city plan in outlying areas. Rome had become a backwater.
There are many good hard copy books on Constantine and the founding of Constantinople -- available at Rome bookstores or through Internet booksellers.
If you want the basics on the net, use the following Constantine and Constantinople links:
Catholic Encyclopedia bio: http://www.newadvent.org/cathen/04295c.htm
Romeweb bio: http://myron.sjsu.edu/romeweb/empcont/e184.htm
Constantine links, Sam Houston State U.: http://www.shsu.edu/~eng_wpf/con-hist.html
Rudolfo Lanciani on the transition from pagan to Christian Rome (an important basic source!): http://www.ukans.edu/history/index/europe/ancient_rome/E/Gazetteer/Places/Europe/Italy/Lazio/Roma/Rome/.Texts/Lanciani/LANPAC/home.html
Remains of the colossal statue of Constantine from the Basilica of Constantine in Rome: http://www.bluffton.edu/~sullivanm/romanconstantine/constantine.html
Constantinople: "From Old to New Rome": http://www.patriarchate.org/ecumenical_patriarchate/chapter_1/old_to_new_Rome.html
An almost contemporary source (Sozomen) on the founding of Constantinople: http://www.fordham.edu/halsall/ancient/sozomen-constantinople1.html
"What, if anything, is a Byzantine?": http://wwwtc.nhmccd.edu/people/crf01/romaion/
Own the commemorative coin!!!: http://www.iccoin.com/romanrombron1.html
P.S.: Constantinople, as everyone knows, simply means City of Constantine (from Greek, Constantinou polis). Istanbul (or 'Stamboul) is a Turkic (not Turkish) pronunciation of Constantinople, not a different name for the city.