Disclaimer: This will be a long answer. I expect that it wont be as good as I want at first. I may make corrections/additions afterwards. I am a native Turkish, drinking kgs of coffee each week and from a coffee addicted family for generations. Therefore, I count myself as a Turkish-coffee hobbyist.
I believe to understand the two or three times boiling mystery of Turkish coffee, you must understand the whole Turkish-style extraction process. I tell you the whole story.
The chemistry is simple: we are extracting some oils from the beans. Two known ways out there, pressure or heat. Turkish extraction is the traditional, genuine, historical heating way. Actually, it is probably the first and easiest way manhood used in history to prepare coffee (I believe both in Africa and then in Ottoman Empire).
When you heat the beans to extract, you need lots of surface to extract efficiently. That's why you grind the beans as fine as possible. The electrical grinders are not always perfect. You may want to use a mortar after the grinding process to be sure if you don't have a Turkish grinder. (This is another topic, I think.)
Then, the aromatic oils you are trying to extract are not of one kind. There are several kinds. The drink you consume under the name of coffee is delicious if it includes many of them. So, you are trying to jail all of them into the water during the brewing process. But, wait... Some of them start to be released from the bean at 75 degrees Celsius, but some others have been already vaporized at 70 degrees Celsius! There must be a way of collecting all of them.
Bingo! You are right. It is the double (triple, quadruple, ...) heating process.
You should heat the water up to some point, extract a subset of the oils, separate them to your fincan (cup). Then repeat the process at a higher degree. Three times is the best practice among Turks. You may define your own temperature slots and repetition count but I think the algorithm is clear now.
Talking about best practice, let me show you a best practice.
Get you cezve (pot). Mine is stainless steel and thick. Your coffee should not boil on the edges first, it must start to boil from the center. (If this is not the case, either your cezve or your fire is problematic.)
If you have a hand grinder, set it to the finest.
Measure the water with fincan (cup) (I prefer soft spring water) and fill your cezve. Traditional fincans are quite thin (in contrast to espresso cups) and porcelain.
After you fill in your cezve with water, add coffee on top. About one teaspoon per fincan is quite OK. (You may change). I don't recommend sugar for any kind of coffee, but it is your taste. If you want to add sugar, it is now or never.
You will see that fine grounded coffee will float and sink in time. Fire up and let the coffee sink by itself. Do not touch or stir. The amount of fire has little or no affect to the taste of Turkish coffee to my experience. There is an urban legend that states that Turkish coffee is best made in wood fire or very slowly in low temperatures, etc. It's nonsense. Then, every aromatic oil will evaporate eventually. You should be quick. These are just romantic/melancholic scenes for old days heard from grandfathers. This was my setting.
After a while, all coffee will eventually sink. This is an indicator. Your coffee is somewhat homogeneously distributed in water. Also you have a nice grease remains already on top of water. However, they cannot form a foam yet. Cause, we didn't stir it.
Now, stop. Please check the foam. You have a quite nice foam already. Very very light color. No bubbles at all. It is not mature yet. If you want to be experienced in Turkish-style coffee brewing, look at that foam a lot. Follow its color and forms.
Stir again. For about a minute. Stirring keeps the foam intact. You may stop from time to time to check the color and then continue. It gets mature when you first see the red/fire color.
Check the second image. Could you see the firefox there? Could you? Ok, now it is time for the first shot. Actually, I would like you to compare the second foam image with the one above. Try to figure out the differences. We started with light color, it turned out to red/fire color. We started with no bubbles, it has bubbles now.
We'll pour some of the coffee to our fincans. I pour the foamy part and fill about 2/5 of my fincans.
Still, I have enough foam and lots of oils to extract in my ground beans. I continue to heat and stir. Don't forget to stir and keep foam intact. This time the foam turns from red/fire to brown/coffee-bean-color. Still, I'm very far from boiling. Fill another 2/5 of your fincans. The fincans are nearly full.
I would like to keep your attention to one point. At this point, you see the famous marbled foam of Turkish coffee for the first time. The reason is, you've mixed two different subsets of oils for the first time. Our final destination is to produce a coffee foam marbled with three colors.
Now, we hardly have any foam left in our cezve. We are heating the coffee for the last shot. This will be very close to the boiling point. You can hardly see any foam now, but mostly bubbles. At one point, the boiling starts in the middle. Take the cezve out of the fire very quickly to prevent burning the coffee and pour the rest of the coffee to the fincans.
(You really need to be quick, so I couldn't find time to focus and burnt the coffee a bit. Still, you can sense the boiling center. Sorry, for this picture and also I cannot provide a triple marbled foam for now as it burnt.)
Normally, there shouldn't be any grounds remain in cezve. Everything must be dissolved homogeneously in water.
Traditionally, we prefer to serve Turkish coffee with half a sip of liquor of sharp tastes. Like mint, quince, sour cherry or bitter almond. This time, it was hazelnut liquor.
A correctly prepared Turkish coffee is not a suspension of coffee grounds in water. Coffee grounds lay at the bottom and never fill in your mouth. The drinkable part stays on top.
Final words: This is it. I hope both theoretic and practical part helps people. Please leave comments so I can further advance this answer in time.