This isn't really my area, but I'll give you the best I can.

Saudi Arabia is actually sometimes known as the Third Saudi State. Saudi here refers to the House of Saud, the ruling dynasty that was established the first Saudi State in central Arabia in 1801. The House of Saud is well known for their alliance with conservative clerics who follow Hanbali fiqh, especially a stream of Salafism inspired by ibn Wahhab (who personally entered a pact with Muhammad ibn Saud, the founder of the first Saudi State). The Ottoman collapse in World War I essentially brought a power vacuum in the region. The Saudis took were able to expand their central Kingdom over most of the peninsula in the next decade. I don't know much of the details, but the Sharifian dynasty, based around the holy cities of Mecca and Medina, were leaders of the Arab Revolt that helped push the Ottomans out (this is all that "Lawrence of Arabia" stuff).

The Sharifians, also known as the Hashemites, initially looked to be the dominant force in the region. The main line ruled Mecca and Medina (the region known as the Hejaz) and the younger brothers became kings of Iraq (initially, he tried to become king of Syria but that failed and Syria) and Jordan. The Iraqi Hashemite line was expelled in the 1950's when Iraq became a Republic, but the Jordanian Hashemite line still rules (Jordan is officially the Hashemite Kingdom of Jordan). The main Sharifian line declared themselves Caliphs once Turkey stripped this title from the Ottomans (the House of Osman), who had claimed the title since the 16th century or so, but used it more frequently from the 18th or 19th centuries as they tried to united Muslims in the face of increasing nationalism. The Saudis defeated Sharifian in the Hejaz in the mid-1920's, shortly after they adopted the Caliph title (the last Ottoman Caliph had apparently supported their claim) but the Saudis chose not to claim the title of Caliph. Since then, no one notable in Sunni Islam (until very recently, as you may have heard) has claimed to be the Caliph.

I don't know the history of the Saudi conquests very well, but the eastern area on the map, along the Persian Gulf, is where most of Saudi Arabia's oil wells are and where most of Saudi Arabia's Shi'a population is (the Kingdom is something like 10-20% Shi'a). The process of Saudi expansion started before the Ottomans left, I should add, and the conventional starting point I believe is the 1902 (re)capture of Riyadh, the capital of the First and Second Saudi States. The expansion really only ended in the early 1930's, around the time the Saudis declared themselves Kings rather than Emirs (and therefore, their state a Kingdom rather than an Emirate). I'm honestly not sure where they got these exact borders from, and names, as I can't quite fit them to historical entities. For example, the Kingdom of Hejaz controlled a lot of what's labeled here "North Arabia" and the Emirate of Jabal Shammar/the House of Rashid--the House of Saud's main rival in central Arabia--controlled some parts of what's labeled here "Eastern Arabia". I think, but am not sure, the Saudis conquered some of these territories directly from the Ottomans. As far as I can tell, including oases of Qatif and Al-Ahsa, then the main population centers and the main historical centers of the Arabian Peninsula's Shi'a population. Historically I believe they had closer ties with nearby cities and states along the Persian Gulf, notably Bahrain, than central Arabia or the Hejaz. Historically, the Ottomans I believe administered the area from the Iraqi (and Shi'a majority) city of Basra, along on the Persian Gulf. Today Damman is one of the three major Saudi cities, but at this point I believe was much smaller and grew mainly with oil (Qatif is now a small part of the Damman Metropolitan Area). I can only assume that Damman was in part developed to avoid the heavily Shi'a areas, but I don't know that for a fact.

Overall, I'm not sure there were ever "five independent regions", so much as two Ottoman regions administered separately from one another (the Hejaz on the West Coast, administered partially indirectly by the Sharif of Mecca, and the East Coast with its large Shi'a population administered I believe from Basra) with central Arabia being fully independent of Ottoman Rule, and contested between the Saudis and Rashidis. I really am not sure what they mean by "North Arabia" and "South Arabia" on this map, whether there was some historical entity in these places that I'm unaware of or they are just marking "border zones" between the Hejaz and the traditional centers of Jordan/Palestine in the North and Yemen in the South but, as I said, I am not expert on this area.

The "two Yemens", I'll just point out, had completely different governments and demographic profiles. Unlike Germany, Korea, and Vietnam, there wasn't really history of "Yemen" being ruled together. Before independence, South Yemen was British and North Yemen was Ottoman. North Yemen, which is where most of the Houthi Shi'a live, was supported in part by Saudi Arabia (though in the North Yemen Civil War between the Royalists and the Republicans, Saudi Arabia had supported the losing Royalists, which again is interesting because the Royalists drew a lot of Shi'a support), and South Yemen had a mostly Sunni population and a Marxist government and was supported in part by the USSR. The whole thing is very confusing. Oh, and the Saudis tried and failed to take over (North) Yemen in the 1930's in the Saudi-Yemeni War (it was the main area in Arabia that was not under British control), but failed. I believe, though, that they settled the border at this point so that the area marked "South Arabia" in this map was determined to be totally Saudi rather than Yemeni. Oh, and I guess I should point out that the Houthis in Yemen (and a little bit over the border into Saudi Arabia) are a completely different sort of Shi'a from those in the Eastern Province (though the ones in the Eastern Province are the same time as in Iraq).