So, there's a lot of stuff in that movie that is definitely incorrect or embellishing events, if not outright racist - but I'm sure you can find loads of movie critiques talking about that, so I'll write a little about the political historical context as you ask. Keep in mind, books upon books have been written on this, so I'm simplifying quite a bit, but I hope to help you along.

It's the First World War. The major players in this theater, the Arab Middle-East, are Britain and the Ottoman Empire, on opposing sides of the war. From their capital in Istanbul the Ottomans rule Anatolia, the Levant, Iraq, and most of the Arabian Peninsula, however their hold is rather tenous in some areas. While Egypt were nominally a vassal of the Ottoman Empire, it was also a british colony. So at the outbreak of war, Egypt are "freed" from the Ottoman sultans. There is fighting in the Sinai and along the Suez Canal, but at first neither side breaks through. The british are also dominant in the Persian Gulf, and there they lead two offensives into Iraq.

What more relevant for the movie, is the Sharif Hussein of Mecca - also an ottoman vassal. He offers the british to revolt, an Arab Revolt. While the Arabs have for the most part been content with Ottoman rule for centuries, this is the age of nationalism, and the Comitee of Union and Progress - also known as the Young Turks - have seized power in Istanbul, and are ruling with a very turk-centric policy. At the same time, many Arabs, especially in Syria and Lebanon are themselves becoming nationalist, following the flowering of Arabic culture known as the Nahda.

Down in the Hejaz, the Sharif Hussein are corresponding with the British High Commissioner of Egypt, Sir Henry McMahon, and are promised British support, both in the war and in the following settlement, most noticeably they promise him to rule an Arab state, encompassing all of modern Syria, Palestine, Lebanon, Jordan, Iraq and Saudi-Arabia. At the same time, there's the Sykes-Picot agreement being made in London, where the British and French decide on splitting the area between them, but that's another story.

So in the summer of 1916, Sharif Hussein declares his revolt. There's some fighting in Mecca, and a long siege of Medina, it seems that not all Arabs are interested in this revolt. Yet the British send their support, among others a young officer named T.E. Lawrence - the titular character. One son of the Sharif, Abdullah, later king of Jordan, commanded one army, while another son Faisal commanded another, in cooperation with Lawrence.

Faisals force were mostly lightly armed, but fast moving beduin cavalry, who would lead a sort of guerilla-like campaign, especially sabotaging the Damascus-Hejaz Railway, that were important for the Ottoman logistics. After General Allenby breaks through Gaza, and occupies most of Palestine, the Ottoman front starts collapsing in 1918.

Following the war, Faisal sets up a Syrian kingdom centered on Damascus, supposed to include all of Greater Syria, yet Palestine is in British hands, while Lebanon and the Syrian coast is in French hands. While Lawrence attempts to talk for Hussein and his sons during the Paris Peace Conference, in the end the Sykes-Picot deal pushes through with a few changes, and the areas are given as mandates to the British and French, with the French defeating Faisals forces and occupying Syria.

There's a lot more to it, but that is the broad lines, feel free to ask follow-ups! If you're interested in the period, A Line in the Sand by James Barr is a gripping read on this period. The Arabs: A History by Eugene Rogan is broader (by 500 years or so) but is probably the best book to get a general overview of the region, and both of them are rather accessible.