The ancient origins of the name
Greece is also called the Hellenic Republic, which refers to the time of Hellenistic Greece between the death of Alexander the Great (356-323 BC) and the Battle of Corinth in 146 BC. This all comes from the Ancient Greek word Hellas, which was the original term to refer to what is now called Greece. Hellas is the word that Hellenistic derived from.
Following Alexander's death his vast empire, which stretched from Greece to Egypt in the south and modern-day Pakistan in the east, was divided into a number of successor states. However, Alexander had helped to spread Greek culture throughout the former Achaemenid (Persian) Empire (550-330 BC).
Over time, especially after the Wars of the Diadochi (332-275 BC) due to the relative peace, great advances were made in many key areas like art, architecture, literature, music, and math. This also led to increased travel and trade, with wonders like the Lighthouse of Alexandria and the Colossus of Rhodes being built.
During this period in history, the cultural influence of the Greeks was at its peak, effecting Europe, parts of North Africa and much of Western Asia. Despite this Greece itself fell off in importance, as the great centers of Hellenistic culture were Alexandria and Antioch, both located in Egypt. Other important cities of the time were located in modern-day Turkey (Pergamon and Ephesus), the Mediterranean (Rhodes) or modern-day Iraq (Seleucia).
The conquest of Greece under foreign rule started with the Macedonian Wars (214-148 BC). This saw the Roman Republic (509-27 BC) and their Greek Allies get into a series of clashes with the major Greek kingdoms of Macedonia (808-168, 150-148 BC), the Achaean League (280-146 BC), the Seleucid Empire (312-63 BC) and the Odrysian Kingdom (480 BC - 46 AD).
Two years later, the Achaean War (146 BC) took place. This saw the Roman Republic decisively defeat the Achaean league at the Battle of Corinth and truly marked the start of Roman domination over Greece. Meanwhile, the Hellenistic period would end with the defeat of Ptolemaic Egypt (305-30 BC) as part of the Final War of the Roman Republic (32-30 BC).
The path to the modern use of the name for Greece
Following the Battle of Corinth, Greece was under the control of the Roman Republic from 146 BC until 27 BC, when it became the Roman Empire (27 BC-395 AD). In 285, Emperor Diocletian (244-311) divided the empire into eastern and western halves. However, it was not until the death of Emperor Theodosius I (347-395) the that East-West divide was made final.
Greece was then ruled by the Western Roman Empire (285-476), which became known as the Byzantine Empire (330-1453). In 1453, Constantinople fell to the Ottoman Empire (1299-1923), starting the period of Ottoman rule over Greece. This was finalized over the next few decades with the defeat of the Despotate of Morea (1349-1460) and the second reestablishment of the Despotate of Epirus (1356-1479).
However, the Ottoman Empire started to weaken in the early 19th century. At the same time a yearning for independence, a major felling of Greek nationalism and a growing economy spurned Greeks into action. This led to the Greek War of Independence (1821-32), which was started by Alexander Ypsilantis (1792-1828) crossing into Ottoman controlled Moldavia and caused sporadic revolts to break out over the Greek peninsula soon after.
This led to the First Hellenic Republic (1822-32) being founded, the first time in modern Greek history the country was not ruled by a monarchy. Later on, the Second Hellenic Republic would be founded, lasting from 1924-35. Greece is currently on its Third Hellenic Republic. It was established in 1974 following the Greek military junta (1967-74) ruling the country after the overthrow of the monarchy and the end of the Kingdom of Greece (1832-1924, 1935-73).