Okay, a point of nomenclature before I begin: you're thinking of military monastic orders. It's a mouthful, I know, but it's an extremely specific term whose meaning is self-evident. A "knight order" (which I take to mean knightly or chivalric order) is not at all like a military monastic order.

To demonstrate just how different the two things are: a knightly order is the Order of the Garter while a military monastic order would be the Knights of St. John. The Order of the Garter is essentially an honor that the Queen bestows upon people, to over-simplify the process. The Knights of St. John, on the other hand, I'll be talking about in more detail.

So let's talk about independence first. To begin with, every military monastic order worth its salt had its own independent land holdings. These holdings may well have been in other countries, but they were functionally independent. They have a bunch of names but "commandery" is the most common. Essentially, a commandery is a piece of land in another state that is owned by a military monastic order which can be used by the order to produce things (or revenue) for its own use, without the country that the commandery is physically located in getting involved. It's a weird situation and kind of like an early example of extraterritoriality. For a modern counterpart, think of a country's embassy in a foreign land. Despite physical location, that embassy is often considered the sovereign soil of the country that operates it. Commanderies were kind of like that.

So, in a sense, most of the major military monastic orders were independent in the sense that they were not vassals to a king. There are exceptions here. Sometimes, as in the case of the Duchy of Savoy, one of these orders would become hereditary with the local ruler inheriting the post of grandmaster (which is what happened to the Order of St. Lazarus before it essentially fractured into three competing branches -- one Savoyard, one French, and one folded into the Knights of St. John).

In other cases we have orders that, despite being legally independent, essentially functioned as an extension of the local state's will. The Order of Calatrava is a great example of this. This particular order was founded in what is now Spain with the express goal of facilitating the Reconquista. This meant that the Order of Calatrava was a regular feature on many of the major battlefields of the period and, during time of peace, it was responsible for frontier fortifications and the like. Grandmasters of Calatrava (who were essentially primus inter pares of the other Iberian orders of the time, such as Alcantara, Aviz, and the Order of Christ -- the Portuguese branch of the Knights Templar that survived the purges) often served at the whim of the Spanish monarchies.

And, finally, there are the truly independent orders. There are really only two: the Knights of St. John and the German Order (as they are known in German; they're known in English as the Teutonic Knights). The Swordbrethren of Livonia were similarly independent but were folded into the German Order (at which point they became a weird vassal state/ally and were known as the Livonian Order). These two (or three, if you prefer) orders were truly independent -- whereas other orders were knights with a few castles spread across Europe, these orders genuinely owned land. the Knights of St. John didn't physically control much (first Rhodes and then Malta with a few short-term conquests, like Tripoli), but the German Order controlled huge tracts of land in Poland and what are now the Baltic States. They still owned commanderies in other European states (the German Order's recruitment was mostly done within the Holy Roman Empire at these locations), but the bulk of their revenue and power came from the land they directly controlled. Or, in the case of the Knights of St. John, the seas they controlled (the Knights of St. John would eventually become well-known as exceptionally good at naval warfare -- or, if you're on the other side of the religious divide, as pirate-knights who had no qualms taking Jews and Muslims as slaves). These orders were truly independent in every sense of the word. They paid obeisance to the Pope, as their spiritually leader, but they often had the authority to appoint their own knight-brethren as de facto priests and, if so inclined, could wage war without the Pope's involvement.

So, the Knights Templar. They relied on commanderies for their physical presence and recruitment (and, upon their dissolution, these would mostly be transferred to the Knights of St. John), but their wealth came from -- get this -- banking. If you were to deposit money at a Templar commandery in France, you would receive what amounted to a receipt and could then give that receipt to a Templar commandery in the Holy Land and essentially withdraw your money. And they also offered loans to states. It was probably these loans that ultimately resulted in their demise: usury never sat well with the Popes and the huge quantities of money that Philip IV owed them certainly didn't endear them to him. And so, when the two ultimately worked together to disband the order on trumped-up charges, we have one of the more interesting events in the history of chivalry -- and an event which, as you probably know, authors have been milking for shoddy thrillers ever since.