Semitic people probably first moved into the Arabian Peninsula around 2000 BC, coming from Mesopotamia. They were nomads when they arrived, who travelled around with their sheep and goats pasturing them in different pastures at different times of year. And they stayed nomads: many of them are nomads today.
In the southern part of the peninsula, on the other hand, the people were farmers. We're not sure where they came from, but the Queen of Sheba mentioned in the Bible may be one of these people.

By the time of Alexander the Great, we start to know a little more about the Arabs, because the Greeks were trading with them. The Romans also traded with the Arabs, who got spices and other things from India and sold them to the Romans for gold.

In the long war between the Sassanids and the Romans, different tribes of Arabs fought on each side. In this Late Antique period, the kingdom of Saba (Sheba) fell apart.

The Prophet Mohammed was born in the northern Arabian trading city of Mecca between 570 and 580 AD. When he was forty years old, he heard the angel Gabriel speaking to him and telling him to start a new religion: Islam. After a slow start, Mohammed made a lot of converts to his new religion, and they began attacking other Arabian tribes to convert them to Islam. After they had done that, they attacked first the Romans and then the Sassanids to convert them.  By 640 (after the death of Mohammed) the Arabs controlled most of Western Asia, and soon after that, under the rule of the Umayyad caliphs, they conquered Egypt. By 711, the Arabs controlled all of Western Asia except Turkey (which was still part of the Roman Empire), and all of the southern Mediterranean: Egypt, Libya, Tunisia, Algeria, and Morocco, and also most of Spain.

By 800 AD, however, under the rule of the Abbasid caliphs, the Islamic Empire had already begun to break down into many smaller kingdoms or caliphates.  The main part of it was ruled from Baghdad in modern Iraq. In the 900's control of Baghdad was taken over by the newly arrived Turks or Seljuks, and the Fatimid dynasty took over Egypt and Syria. In the east, the Ghaznavids took over Afghanistan and then northern India about 1000 AD. 

In 1096 AD, the Crusaders arrived from Europe and conquered a good deal of Israel and Lebanon from the Fatimids. But by 1200 most of the Crusaders' land had been reclaimed by the Mamluks and the Ayyubids (under Saladin).
 

In 1260, the Mongols invaded West Asia, and conquered the eastern part of the Islamic Empire, as well as northern India and Afghanistan. During this time also, Europeans gradually pushed the Arabs out of Spain (the Reconquista), finishing up in 1492 AD. 

By 1453 AD the Ottomans (successors to the Seljuks) had begun to establish the Ottoman Empire by conquering Constantinople (modern Istanbul). In 1517, they conquered Syria and Egypt, and by 1639 AD the Ottomans had taken Iraq. The Ottoman Empire lasted until the end of World War I in 1918 AD, and takes us out of the period covered on this site.

Source: History for Kids
http://www.historyforkids.org/learn/islam/history/history.htm