This is hard to answer for three reasons. First, "when it first started" is not quite a clear time period, so there are multiple reasonable answers. Second, records just don't really exist of this, since no one was taking a census at the time, so we have to do a lot of estimating based on scanty information. Third, the Mongols were nomadic and lived in a clan-based society where allegiances constantly shifted, making it more or less impossible to pin down how many people actually belonged to any particular group at any particular time.

Because of all that, I'll give you three answers corresponding to different time periods: during the time of Yesugei, either 50-200 people, around 2,000 people, or about 200,000 people, depending on how you count. During Hoelun's exile, 10 people. At the 1206 Kurultai, around 2,000,000 people.

Temujin, better known as Chinggis Khan (and even better known, wrongly, as Genghis), derived his authority from being the son of Yesugei, who was the de facto leader of a Mongol confederacy around 1150 AD. Yesugei was descended of people who were officially recognized as Khagans and uniters of the Mongols, so Temujin was roughly equivalent to a prince. Yesugei's clan, in which Temujin was brought up, was in a sense the royal camp, so it would have been a bit larger than others at the time.

Now, Yesugei's camp would have been made up of essentially two parts. One would be his immediate kin, primarily brothers/sisters, wives, concubines, and children like Temujin. Also included in this would be some servants to keep things running. The second part would be other, similar groups that followed his camp around.

Sources, as I said, are vague, but there's one we can use to reasonable effect here. In the Secret History of the Mongols, there's a discussion of who remained loyal to Hoelun (Yesugei's wife, Temujin's mother) after the death of Yesugei. At this point, almost everyone who had belonged to Yesugei's camp had deserted to follow a new leader, leaving only Hoelun and her immediate kin. In my translation, it reads:

Before long Hoelun had no people left, except her immediate household. There were her sons Temujin and Kasar, and her two other sons and her daughter, but these last three were still very young. There were one or two other widows, of Yesugei's men, and their infants; and two boys who were half-brothers of Temujin. These two, Bektair and Belgutai, were Yesugei's sons by some other woman, before he met Hoelun; they had no kinsmen, and Hoelun had brought them up with her own sons. The mother was part of her retinue. Then there were a few servants, who were not not much good for anything, except perhaps the old woman Khoakchin, who was at least shrewd and energetic. Any serving man who was capable and loyal would sooner or later be granted weapons, but Hoelun had none who were fitted for it.

So doing the math here, we have somewhere between 15-20 people, which as I said represents close to a minimum. In a healthier time, when Yesugei was alive and respected by all, a core of 50 to maybe 200 would be reasonable: a leader would have more wives and concubines, and therefore more children, all of which would require more servants, some of whom might be primarily tasked with bodyguard duty. Around that core would be up to a couple thousand people, all hanging around the leader for the various reasons that people have always congregated to palaces and the sides of kings. So when Yesugei's camp would go on the move from one grazing area to another, an observer would have seen a train of a few thousand men, women, and children, along with a great deal more animals being driven along.

However, at the time, Yesugei was not just the leader of this camp, but also the acknowledged leader of all the Mongols. This is where it's really impossible to pin down a number, because being the leader meant essentially that when he called groups to arms, others would listen. But that loyalty was conditional on what the call to arms was for, how much plunder could be had from the battle, was there anything else to do, was there someone that you liked more who was committing manpower to the other side or staying neutral, etc. We know that Toghrul, who was Khan of the Keraites (another tribe like the Mongols), offered Temujin 20,000 troops for a war against the Merkits. Jamukha, his great friend and rival (seriously there should be a movie about this relationship) attacked Temujin in 1187 with 30,000 troops.

We can use these numbers for estimates. These forces would have been substantial, if not the entirety of a leaders' manpower then something close to it. If we say that fighting-age men were 1/3 of all men, and that there were roughly an equal number of women, we would say 20,000 fighters corresponds to 120,000 population. 30,000 corresponds to 180,000 in population. A powerful Khan in a good situation likely could have drawn on something like 200,000 people.

Finally, the 1206 Kurultai. This is the point where Temujin has unified not just the Mongols, but all the steppe peoples, under his command. Here, he is named Khan, and takes the name Chinggis Khan. While he did not take the title of Khagan (one who rules a khaganate, an empire), this is effectively the "Declaration of Independence" moment for the Mongol Empire.

Again, figuring out a population here is a lot of guessing. Start from that 200,000 figure for a clan. We can name quite a few clans besides the Mongols--they're all mentioned in the Secret History--like the Keraits, Naimans, Tatars (which is why the Russians called the Mongol period "the Tatar Yoke" and why Europeans called them "Tartars"), Buryats, Merkits, Uyghurs, etc. These large clans would have included some smaller clans who attached to them for political reasons, so there would have been also a few smaller and unaffiliated clans. However, this doesn't move the needle all that much. That Kurultai would have represented a total population of somewhere in the neighborhood of 2,000,000 people.