The short and perhaps disappointing answer is that it varied depending on the period.

The practice started during the reign of Murad I, who came to the idea of islamizing and reeducating the children war prisoners after the sudden expansion of the empire. For that purpose, he formed the Acemi Oğlanlar Ocağı -- its goal was teaching the kids to read, write, be proper Muslims and fight. At first they were only recruited from the borderlands, where their akıncılar troops would raid the enemies' territory and take the captured kids with them. 1/5 of the kids would be taken to the Acemi Oğlanlar Ocağı and they'd become pencik oğlani, roughly translated kids of the fifth, and subordinate to the Yeniçeri Ocağı. According to Tevarih-i Al-i-Osman, this practice already existed in 1360., and according to Tacü't-tevarih and Oruç Bey, it began in 1362. I won't go into details about this, but these years are approximately the early days of the practice.

After the battle of Ankara, the conquests slowed down, and so did the influx of war prisoners, so Murad II issued the Devşirme Kanun which made it possible to "take" kids from the territories of the Ottoman Empire. They were called devşirme oğlanı. The goal of this, of course, was strengthening the Janissaries, and while it didn't completely replace pencik oğlani, it turned out to be more practical so it became more prevalent. It started in Rumeli (basically, Balkans), specifically the area of Albania, Macedonia, Greece and Bulgaria, and then spread to Serbia, Bosnia and Herzegovina and Hungary. Only around the end of the 15th/beginning of the 16th century does it spread to Christians of Anatolia.

The shaky consensus is that at first the children were recruited from all Christians indiscriminately, whereas gradually it was limited to Albanians, Greeks, Bulgarians and, most likely, Armenians. We know for a fact that kidnappings from certain nationalities was prohibited later on. Additionally, according to Aşıkpaşazade, Janissaries were recruited among friendly Christian communities, and the choice to join full corps was voluntary (gotta do some fact checking for this, don't take my word for it) but the frequent attempts to sabotage Ottoman authorities' efforts suggest otherwise. However, there are indeed also some examples of parents bribing the Ottomans to take their kids because the blood tax opened many opportunities in terms of social mobility. Likewise, there's a claim (Müriü't-tevârih and Kavanin-i Yeniçeriyan) that the Bosnian and Albanian Muslims asked the authorities for their children to be taken away. This was during the reign of Mehmed the Conqueror. And there is indeed one document from 1666. that orders that they can be blood taxed only if it's voluntary and if the children are 15-20 (well, not really children at that point).

Now, a frequently asked question is: How often were these children levied?

According to Benedikt Kuripesic, it was every year. Nicolas de Nicolay and Gabriel de Luetz , on the other hand, say that it was every four years. Beneditto Ramberti (very well informed) agrees with them, and also says it was every four years. Various other authors including Marco Pigafetti (most prevalent opinion, can't bother listing) says it was every five years, while some go as far as seven. This doesn't necessarily mean that any, let alone most of these were wrong. Frequency of this tax most likely depended on the location and the needs of the empire. François Pétis de la Croix mentions that peasants would often move to the cities to avoid this "brutal and shameful tax", but let's not exclude the possibility of artistic exaggerations. Also, let's not forget that they wouldn't tax the same villages over and over again so as not to overburden them and cause possible uprisings and general unrests.

Another FAQ with an equally ambiguous answer: How old were the kids?

During Mehmed I and Murad II's reigns, they were between 7 and 8. In 1554. they were 10-12. Another traveler from 1640/1. says that they were 12-18, so there's a notable increase over time. A ferman from 1622. demands that the children are 15-20, and in 1638. 16-25. The explanation for this is that the military needs were high, and there was no time to wait for kids to grow up. Ironically, while the recruitment was more voluntary (for obvious reasons), the troops were far less fanatical as you're recruiting formed individuals.

So, how did they actually get recruited?

Until the first half of the 16th century, the recruitment was primarily the duty of local authorities -- beylerbeys, sancakbeys and kadis. But these guys were essentially corrupt as fuck, so the Yeniçeri Ocağı itself took over that function. Devşirme memuru (the guy who was responsible for the tax) would get a ferman and a letter from the Janissary ağa, which'd contain instructions like: How many kids? From what village? How old? The average was that every 40 houses would have to give one kid, but it varied. The janissary Konstantin Mihailović calls these kids cilik which is probably from Persian 1/40, and there's an edict from the 16th century demanding a child for 1/40 households. However, I've also seen sources mentioning 10th children. I can't remember it right now. Melhior Sejdlic in 1556-1559. claims that five children were taken per village. Constantino Garzoni also notes that the authorities would never take away only sons, which is confirmed in the ferman of 1666., albeit more as a recommendation than an order. It also orders not to take away more than one child per household, not to take away children who are ugly/dumb/fat/short/etc, who are circumcized, who are married, who know a trade, live in a city or speak Turkish. It was believed that city kids are not good soldiers. Kids from parents with good reputation were preferred, and kids from the local priests were not spared either. Locals would also have to pay kaput bedeli, which is a tax for the clothes, as well as a tax for taking the kids to Constantinople, plus some cash for the local authorities (sometimes this was forbidden). Whoever wants to know more might want to read Evliya Çelebi. Of course, there were all sorts of measures to minimize cheating, so if anyone's interested I can elaborate. I might also translate some documents related to the practice if I have time tomorrow.

After 100 - 200 kids were collected, they'd form a group. Sometimes parents would bribe the sürücü, the guy(s) responsible for taking care of them on their way to Istanbul, to swap their kid with some random one, so this bribery was strictly punished later on. The group would be supplied by random villages along the way, and this was usually quite a burden on them so they'd often try to resist it by not letting the caravan stop by. That's how the village of Breziste attacked a caravan around 1565/6. They'd travel by foot or on horses and the journey would last a few months depending on the location.

Was everyone a possible target of the tax?

Absolutely not. Aside from the major cities, which I've mentioned, many ethnic groups and areas were excluded:

• children that lived mixed with Turks, Georgians and Kurds, and who lived between Karaman and Erzerum

• no kids from Croatia, Central Hungary and Belgrade, because Hungarians and Croats were unreliable and would switch back to Christianity at first opportunity (Kavanin-i Yeniçeriyan)

• no mentions of devshirme in Wallachia and some other parts of modern day Romania

• no Arabic and Kurdish kids, nor Jewish kids whose parents were merchants, some also claim no Armenians either but this is shaky

Some parts of rayah were excluded if they took up some military or half-military duty, usually in border areas, in exchange for privileges (first and foremost ease of various tax burdens). This privilege however was occasionally abolished and varied across the Empire. Again, I can elaborate if somebody's interested, but there are too many of them to list here.

When did the blood tax end?

The commonly accepted answer for a while was that Murad IV ended it in 1638. However, there's some evidence (a few documents and the testimony of François de La Boullaye-Le Gouz from 1653) that it continued in mid 17th century. Evliya Çelebi, some Venetian sources as well as Matej Gondola from Dubrovnik suggest that it continued throughout the 17th century, and likely a part of 18th until Ahmet III.

Edit: grammar and typos