Mare Nostrum literally means "Our Sea" in Latin. In context, it usually relates to the time when the Romans were able to call the Mediterranean their sea. By 117 AD, at its' height, the Roman Empire controlled all the lands surrounding the Mediterranean Sea. From the northern coast of Africa into Syria and Asia Minor. All throughout Macedonia, Gaul, Spain, Britain and obviously Italy. All the islands throughout the Mediterranean ranging from Cyprus, Crete, Sicily, Sardinia and Corsica were also under Roman authority. The strength of these words most likely brought pride to anyone of the time that could call them self a Roman.
Military: After the 3rd Punic War, Rome could easily be considered the greatest military power in the Mediterranean, controlling at least eighty percent of the land surrounding the Mediterranean along with an army that had not only defeated the Carthaginians in the Punic Wars (such as in the Battle of Zama), but would also move on to defeat the tribes in Gaul and many other regions. This army would later evolve into a standing army, reaching an all time high of over 350,000 men.
Religious: The spread of Christianity of the time was based on the Mediterranean Sea, which was at the center of the Roman Empire, allowing for safe passage of missionaries to spread the word of God throughout the empire. In one case, a missionary by the name of Paul, a Jewish Missionary from Asia Minor, set up churches around the Mediterranean. Paul's' main source of transportation during his 4 journeys was through the Mediterranean to preach the word of God to all.
Economic: Economically, Rome controlled the Mediterranean in many different aspects. With the Mediterranean under control of one nation (Rome), trade networks flourished allowing for both the spread of goods and ideas to Roman towns and city-states throughout the empire. The military strength of Rome which allowed for growth all around the Mediterranean also brought new wealths to the Empire of Rome. This is because with conquest of other nations, came more land along with slaves to work the land. One outcome of conquest all around the Mediterranean was wealthy Roman families buying large estates known as latifundias. These latifundias were then worked on by the people conquered. Another factor which proved Romes power in the Mediterranean was the fact that it controlled "bread-basket" provinces which brought in goods such as grain into the rest of the empire, especially Rome (being because Rome had a deficiency in grain production and needed other sources of grain).
Political: Politically speaking, as stated, Rome controlled many different regions which surrounded the Mediterranean. At times, the Roman Government would set-up "puppet leaders/monarchs." These regions would be considered provinces of Rome (ie Egypt). This spread Roman influence around the Mediterranean, bringing Romans to a point where they could call the Mediterranean their sea-- Mare Nostrum.
Other references to Mare Nostrum include:
MareNostrum Supercomputer: Greatest supercomputer in Europe
The Board game: Serge Laget's board game created in 2003
The Novel: Spanish novel written by Vicente Blasco Ibáñez
The Film: Silent 1926 screenplay.
The Watch: a prototype watch designed for officers in the Italian Navy in 1943.
The Music: MareNostrum being a musical selection from the documentary TV series Victory at Sea, explaining naval warfare in World War II. The music was composed by Richard Rogers.
The rest was from my knowledge of the Roman Empire.