Yet there is one of the Æsir who is called Týr: he is most daring and best in stoutness of heart, and he has much authority over victory in battle; it is good for men of valor to invoke him. It is a proverb, that one is Týr-valiant who surpasses other warriors and does not waver. He is wise, so that it is also said, that the one that is wisest is Týr-prudent.
--Gylfaginning, The Prose Edda by Snorri Sturluson, ca. early 13th century
Týr, Sam & Steph Braithwaite, 2014 for Hero's Journey
Týr is the god of... well, actually, there lies the controversy. You see, Týr is definitely a very important Norse deity; he's in all the stories and shows up in worship, attestations by other cultures coming into contact with the religion, and all over artwork from the region. He's clearly respected and well-liked, appearing as a heroic figure in both the Poetic Edda and Prose Edda as well as in snippets elsewhere. He's related to lots of other important gods, and he's got his own spotlight Ragnarök death-battle.
But Norse mythology is a lot looser about the "god of X" designations that a lot of other religions like to use. You can loosely call some of them a god of a particular concept or area, but they're pretty much all more associated with stories about civilization and humanity and less directly with specific spheres of influence. It's much harder to say "god of love" or "god of war" or whatever for this particular religion's deities... and nobody is harder to label than Týr.
So Týr defies categorization, but while on the one hand that's just what's happening in the Norse pantheon, on the other hand he specifically has gone through a lot of changes and redesignations over the centuries (or at least, we assume so; scholars love to argue about Germanic subjects and this guy is no exception). Týr has been variously associated with bravery and courage, warriors and combat, justice and law, and even animal husbandry. The Romans conflated him with their war god Mars, suggesting that they thought that his most important attributes were warlike ones, but then again the interpretatio romana is not exactly famous for sensitive and thoughtful collaborations with other religions - in fact, some Roman writers also conflated him with Mercurius, god of travel, communication, and commerce. They also sometimes referred to him as "god of the thing", meaning the Norse political and justice body of the þing or assembly of rulers, which also makes it seem like they thought he had a fairly important role in politics and mediation.
You know he's legit because his hat has the tiny horns
Since there isn't much of a written record from the actual practicers of the religion as opposed to the Christian writers recording it for posterity after showing up to convert everyone, it's harder to know what the Norse themselves thought Týr was about. There is definitely a strong element of him as a god of warriors, with surviving poems and spells suggesting to invoke his name to bless weapons for combat, but he also appears again in what seem like more political contexts; Gylfaginning refers to him as so wise that anyone who is especially clever is sometimes called Týr, and Skáldskaparmál uses his name as a kenning for all the other gods, implying that he has some sort of universal importance.
So if Týr's traditional worship and divine function is a little opaque, where else do we turn? Well, to the stories of his exploits, of course, which are few but all very high-octane. The most famous one is his relationship with Fenrir, the terrible monstrous wolf born of Loki and the giantess Angrboða, a creature that grew from the size of a normal wolf to become so huge, powerful, and deadly that all the Æsir were terrified to go anywhere near it. Týr became the wolf's caretaker almost by default, simply because he was the only god brave enough to feed and interact with it.
Who's a good cosmic monster boy? Is it you? Is it you, buddy?
If being a massive dangermonster wasn't enough to distress the gods about Fenrir already, the völva that the king of the Æsir Óðinn resurrected for information prophesied that Fenrir would devour Óðinn at Ragnarök before being slain with great difficulty by his son Víðarr. So it's not very surprising that the Æsir started plotting to get rid of the wolf... which is of course where Týr, the wolf's keeper, comes in.
This is an interesting part of the myth, because it depends on Týr's relationship with Fenrir and on the wolf's trust in him to even try to pull off. After your usual carousel of shenanigans about trying a bunch of wolf containment solutions that didn't work, the Æsir had the ribbon Gleipnir made by the dvergir, the only thing in the universe strong enough to hold the wolf in perpetuity; but as a result of said shenanigans, Fenrir was understandably suspicious that the Æsir were Up to Something and refused to let them put it on him until Týr reassured him that he would be safe and that they would let him go afterward if he couldn't break the bonds himself. Still suspicious, Fenrir asked that Týr put his hand in Fenrir's mouth as insurance.
It's not hard to see where this story is going: the Æsir refuse to release Fenrir once he's bound, and as a result he bites Týr's hand off, a betrayal for a betrayal. Fenrir is relocated to a distant prison where the Æsir keep his mouth pried open with a blade, causing his mingled saliva and blood to run out and create a lake around it so that no one can come near, and Týr goes home minus one limb, and everyone is stuck with that status quo until Fenrir's prophesied escape to wreak havoc at Ragnarök.
Pictured: exactly zero people having a good time
A lot of ink has been spilled over the implications of this story, which has a lot going on under its surface. For one thing, while Fenrir is terrifying and the Æsir are usually treated as the heroes of the story for attempting to keep him from eating things he shouldn't and heading off the apocalypse he'll be part of, he's also undeniably innocent at the time this story takes place; he hasn't actually done anything to anyone except be big and scary-looking in their space, making the Æsir's collective decision to throw him in torture jail forever a betrayal on a pretty big scale (especially because his father Loki, who is conspicuously absent from this story, is one of the Æsir himself). Týr being Fenrir's caretaker and the only god who interacts with him adds another layer to that; while there isn't any detail on what their relationship is like, there's a reason he's the god that Fenrir trusts enough to take his word and allow himself to be bound, so the betrayal is compounded again (perhaps on more than one side - modern writers often wonder if Týr might not have been happy about doing this to his giant wolf charge himself).
And Týr himself has arguably been used as a sacrifice by his own fellow Æsir, and while everyone is definitely impressed by his bravery and positive about him forever after, you have to wonder if he was wanted to make that sacrifice, or if he was hoping that either Fenrir or the Æsir would spare him from it and was betrayed when they didn't. It certainly seems to be a sore subject; when Loki is verbally attacking all the Æsir in Lokasenna, he tells Týr to back off because he doesn't get to be "the right hand of justice" anymore, having lost that hand in his betrayal of Fenrir. It's Týr who says, possibly in argument or possibly in agreement, that he just has to miss his hand but that Loki has to miss his son forever, and after Loki tells him to shut up, he does for the rest of the scene.
Anyway, it's all very heavy and emotional. While Fenrir is busy eating Óðinn at Ragnarök (a self-fulfilling prophecy, as if he didn't have a good reason to want to devour the Æsir before, he definitely does now!), the great terrible dog Garmr from Hel will be the one Týr is fighting, a struggle that ends in both of their deaths. If you find it slightly suspicious that Týr is fighting a giant canine at the end of the world but it's just slightly different from Óðinn doing the same thing, you're right there with plenty of scholars of Norse myth. One theory is that Garmr is one of Fenrir's many descendents, seeking revenge for his progenitor's suffering, while another is that Garmr is Fenrir by another name. Of course, that would make it hard to have him getting killed by two different people at Ragnarök at once... but only if you aren't aware that there are even more onion-like layers to Týr's divine role.
You see, Týr is weird when compared to the other sons of Óðinn. He doesn't hang out with Óðinn much and he doesn't seem to have the same interest in youthful shenanigans they all do. He often appears called upon by worshipers and even the other Æsir as an authority figure, not a subordinate one. His name, image, and stories seem to predate a lot of the other Æsir's, sometimes by a considerable number of years.
But have you considered just sallying forth harder?
The main theory about this is that Týr actually wasn't originally one of the sons of Óðinn. He was possibly originally the leader of the pantheon, or at least an authoritative and powerful figure in his own right. Hymiskviða claims he's the son of the giant Hymir, not Óðinn at all, and his mother and grandmother are referred to as giantesses, the latter with nine hundred heads; he goes forth and meets these people with Thórr and is recognized by both his mother and grandmother when he does. Add to that the fact that Týr's name is used in kennings for Óðinn suspiciously often, and you have a full-blown scholarly conspiracy theory.
Some of Týr's worshipers believe he was - or even is! - the true ruler of the Æsir, a wise and brave figure who stepped down or was pushed aside in favor of Óðinn at some point, while others believe that he and Óðinn are the same person in different guises, not too hard a theory to believe in given that Óðinn is a continual shapeshifter and impersonator. Others treat him as simply the son of Óðinn he is in the modern day, no less important or powerful for being of a lower generation. It doesn't make much difference to his worship, which often involves the sacrifice of doll arms or arms made of food that can be eaten by worshipers in honor of his loss of a hand (or in older times, at least if you listen to the Romans, real arms cut from those slain in battle and hung from trees to honor him). Whatever he once was, he is now the god that his worshipers look to for courage in the face of adversity and protection in times of battle and strife.
In Hero's Journey, Heroes chosen by Týr are expected to show the same courage their patron did and not to flinch from danger, and they often emerge victorious even against great odds. When they spend a Chapter Labor to use a Blessing, they choose one of the non-1 dice resulting and convert it into a 10, and can even choose to do so on command by spending Devotion, allowing them to achieve glorious success when they most need to.
Týr may be confusing and complex, but he's also very simple: a god who represents courage, justice, and mediation, not only by being the authority who dispenses these things, but also by representing the consequences of trying to wield such tools in bad faith.