Nuuttipukki comes from a tradition that is surprisingly documented in essentially every culture from around the globe of a "wild man." The story is essentially about large, vengeful, semi-god-like or at least demi-god-like, beings that live out in the wilderness. They almost always cover their bodies in crazy outfits made of whatever they can find in nature- furs, dirt, feathers, pine needles, leaves, bark, you name it -and almost always wear a terrible mask or headpiece of some sort (often depicting either something scary or something creepy in general). They also almost always have weapons on hand of some sort. Like I said, nearly all cultures have one or more versions of this character somewhere in their history which is extremely interesting, but perhaps even more interesting is their connection to horned animals- in particular, the goat.
Yep, we are back at the ol' Yule Goat. And this is perhaps where the story gets a bit muddled. Since this is extremely ancient history and much of what we know about ancient Europe's history is extremely incomplete, it's hard to know when the character of a goat went from an anthropomorphic goat, to a man dressed creepily as a goat. What I mean is, we obviously have well documented evidence of people dressing up as a human-sized walking and talking goat who visited houses. But at some point- and it's hard to tell if that point was before or after the documented Yule Goat practices, though it's likely before -that character was not meant to be a goat at all, but was supposed to be a man-creatures dressed in the guise of a goat. The Nuuttipukki.
This guy is ultra super creepy. He wears clothes made from all the natural materials I mentioned above, generally has horns on his costume, usually carries a switch for whipping (or a worse weapon for serious devastation) and often a sack, or chains, or anything else that one might expect a haunted house character to carry. Traditionally his face is covered with a mask, which is always super creepy and most generally made of one of the three following options: inside-out animal hide, paper, or tree bark. Seriously this guy looks like a serial killer. If we took the Belsnickel tradition and mixed it with the Yule Goat, and gave it all of the malice of the Krampus, that's what this guy would be- which makes it easy to see where all of these weird traditions spawned.
Pagans used to have festivities to honor the return of the sun for Yule, this monster is thought by some to be an ugly creature and frightened children while others believe it was an invisible creature that helped prepare for Yule. Most theorists believe when Christianity began incorporating Pagan ways into their festivals in order to justify the action, they merged the Pagan figure with an already existing Catholic legend known as Saint Nicholas to create Santa Claus. Initially he would visit houses demanding mead. Though he was feared by the inhabitants, they would let him in and feed him, lest they be killed. The tradition spawned from this was that young men would dress as this guy and run around the neighborhood demanding alcohol. This eventually turned into him visiting houses and performing dances in exchange for food, which eventually turned into treats of any sort, and has finally turned into leaving treats (like cookies) for which he would leave treats, toys, and other presents. Eventually this guy became most well known as the Joulupukki- after being strongly associated with Yule - which literally translates to Yule Goat. The word "pukki" comes from the Teutonic root "bock," which is a cognate of the English "buck", "Puck", and means "billy-goat." So he's the Christmas Goat Man.
Joulupukki is said to live and work in Korvatunturi, in Lapland. A prominent animal species in Lapland are the reindeer, which is where the concept of Santa's reindeer was born from as the story became less about a horrible monster and more in line with guys like Belsnickel and Santa. His assistants are called tonttu, or more precisely joulutonttu (from Swedish tomte); they are not elves, but are essentially human, often dwarflike in character, with long white beards. In later developments, the joulutonttu also ride goats. It is easy to see how this tradition infiltrated other tomte/Yule Lad traditions and has eventually been interpreted popularly today as Santa's Elves.
Even Joulupukki himself has changed much from his original form. He still exists in Finland and is even still called Joulupukki, but no longer is he a monster or a wild man. Though still carrying his Goat moniker, he now dresses and behaves exactly like Santa Claus. In fact, the US-based Coca-Cola Santa Claus was designed by the son of Finnish emigrants, Haddon Sundblom. So without this thing, we would surprisingly not have the Santa we all know and sometimes love today. He has gone a long way from dark mysterious monster to popular commercial icon.