"Slave soldiers" like the mamluks were a common phenomenon in Islamdom because they solved a multifaceted problem that tended to face rulers in this region moreso than in Europe, namely, if you preside over a region in which you cannot reliably depend upon the population you rule to fight for you (typically because they speak a different language, or come from a different tribe, or have divided loyalties with other factions in your domain, etc) then how do you ensure that you have a large, powerful and loyal military force.

Slave soldiers were the solution to this, as first practiced by the Abbasid caliphs who would enslave entire Turkic tribes in order to create a warrior class that was, in theory, directly loyal to the caliph and no one else. The caliph fed them, clothed them, housed them, provided them women and plunder and so on and so forth.

The Mamluks, much like the Ottoman Jannisaries, were a more systematized form of this (the word "mamluk" itself means "owned" i.e. a slave) whereby young boys were purchased as slaves and brought up as part of a military regiment or household. These slave soldiers were given incredible power as the military component and leadership of the military of the state. While their children could not, themselves, be mamluks they often became administrators such that by the 13th century the mamluks and their kin were in effective control of both the civil and military portions of the government.

Now, to go back to the takeover. This idea of absolute loyalty to the sultan (or the Caliph, or whomever) works great-- while the ruler is alive. But when that ruler dies, and is succeeded by someone to whom the slave soldiers have no loyalty, or if they are succeeded by a child, then you end up with a very week head of state who is supposed to command the loyalty of a very effective, merit oriented martial class.

The result was that in many of the instances where this system was tried, the slaves eventually became the masters. While the Abbasid caliphs were nominally the heads of state over their territories until 1258 they had effectively lost all real power to turkic slave soldier Emirs (lit. commanders, but effectively regents) from around the time of the Anarchy at Samarra several centuries earlier.

In the case of the Mamluk's, the Egyptian Ayyubid dynasty, founded and held together by the brilliance of Saladdin, was pretty mediocre after his death. While Saladdin's army was diverse and did include Mamluks, the Mamluks were increasingly relied on as the exclusive military force of the state, particularly under as-Salih who had taken power in a dynastic struggle and (surprise!) had to rely on slave soldiers because as ethnic Kurd (like all the Ayyubids) with dynastic rivals he needed a military that was loyal to him.

All of that was fine until as-Salih died, and his apparently quite useless son was made ruler, all during a time of crisis when Crusaders had briefly seized Damietta in the Seventh Crusade in 1249. Under the leadership of the extremely talented General Baibars, the Mamluks murdered as-Salih's son, Turanshah, kept a titular Ayyubid relative as the nominal head of state but put themselves in power (edit 2: Baibars was a powerful Mamluk at the time Turanshah was murdered but his rule came later.)

What's interesting about the Mamluk model though is that it was basically self-perpetuating. It turned out that although a hereditary dynasty was capable of recruiting slave soldiers, so were powerful slave soldiers themselves. As such, while the Bahri Mamluks are often referred to as a "dynasty" they did not have hereditary succession, but rather just kept the original institution going of its own accord only now loyal to themselves as Sultans rather than to a hereditary dynasty.

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As for being like the Samurai, unfortunately I'm not very familiar with the Samurai as a class, but I can think of a couple points of difference and one potential similarity. It's my understanding that the Samurai were exclusively the only legitimate soldiers in Japan, the only ones allowed to own swords. That was not the case with the Mamluks, who certainly relied on other levies and professional free-soldiers who formed a part of the Egyptian army. But the upper echelons of power were reserved for slaves. Also as I've mentioned this was not a hereditary class, and in theory (though less so in practice) actually could not be inherited, whereas its my understanding that Samurai status was hereditary.

That being said, I suppose you could make an analogy, less so with the Mamluks who seized power than with the Abbasid caliphs, between the Shogun and the Emperor, wherein the Shogun holds all of the actual power and the Emperor was essentially a figurehead.

Again, I'm sure a Japanese scholar will tell me I'm full of it, as this kind of comparison is not something I've studied.

Edit: typo Edit 2: mistaken detail over Baibars, above.

Not an answer but a follow up question : For how long did the Mamluks rule over Egypt? And did they expand to areas west of Egypt? (e.g. Tripoli, Tunis, Algiers, etc.)