The "old Turk" refers to the deposed Turkish sultan, Achmet III, who Candide meets in Chapter 26. It is clear that Achmet, and the other five former kings and rulers with whom Candide finds him, is used by Voltaire to satirise the aristorcracy, and in particular, the feeling they have that it is their "right" to rule. Note how Achmet III introduces himself to Candide:
I was Grand Sultan for many years; I dethroned my brother, my nephew dethroned me, my viziers lost their heads, and I am condemned to end my days in the old seraglio. My nephew, the Grand Sultan Mahomet, gives me permission to travel sometimes for my health, and I am come to spend the Carnival at Venice.
Although figures such as this deposed Sultan and the other kings believe that they possess the right to rule because of their blood and lineage, this belief is mocked by the way that all of them, the Sultan included, testify to the way that they have been deposed and have deposed others. In addition, the way that they are former kings and rulers highlights the massive impact that capitalism was having on society at the time and how older systems of thought and belief such as the aristocracy's divine right to rule were becoming increasingly scrutinised and questioned. This deposted Turkish Sultan is therefore used by Voltaire to comment on some of the ridiculous beliefs held by people in his society, specifically regarding the idea that the aristocracy were intended by God to rule.
The Old Turk appears in the final chapter of Candide. At the end of all of their travails, Candide and Pangloss encounter a old man who invites him to dine with him and his family. Candide and his companions are struck by how happy the man is, even more so than kings and great men they have been around. The men realize that happiness consists in the ability to handle one's own circumstance. The old man is oblivious to the politics and great philosophical debates of the day, but this does not bother him. He and his family eat well, live comfortably, and genuinely love each other. From this man, Candide derives an insight: "We must cultivate our garden."
The Turk represents the value of minding one's own business, taking care of one's responsibilities, and caring for each other. All of these things will bring one much more happy and productive times than the kind of metaphysical speculation Voltaire uses Pangloss to represent.