In The Lord of the Rings, Aragorn is Isildur's heir and the rightful king to the throne of Gondor. He guides the hobbits (and the One Ring) out of Bree to Rivendell, where the council is held to discuss what should be done with the Ring. Aragorn also volunteers to join the Fellowship of the Ring. When the Fellowship breaks, Aragorn ends up traveling with Legolas and Gimli to Rohan and later to Minas Tirith.
Ultimately, after the War of the Ring, he is crowned and marries Arwen, having proven himself worthy. Throughout the series, Aragorn proves without a doubt that he is a badass. Here are ten of those moments.
Paths of Dead
While in the Paths of the Dead, Aragorn calls for the dead army to assist him in the final battle against Sauron. These ghosts had lived in the past since Isildur's day when they refused to fight for either his or the enemy's side and he cursed them to stay until one of his heirs called upon them to fight Sauron.
Aragorn must prove that he is Isildur's heir, a legacy he had been reluctant to accept previously, in order to win their alliance.
Helmet Kick After Hobbits Gone
In The Two Towers, Aragorn, Legolas, and Gimli are on the hunt for Merry and Pippin, two of the hobbits, after the group has split up. After finding what seems to be their dead bodies, Aragorn kicks an Uruk-hai helmet in frustration.
Interestingly enough, Viggo Mortenson broke two toes during this shot, and the realness of it (as well as the scream) was what convinced Peter Jackson to use it in the finished film. Mortenson's impulsive decision led to a great shot, and also a badass moment for his character.
King of Gondor
Gandalf crowns Aragorn King of Gondor at the end of the series. It's only after this that Arwen is presented to him as his bride. This whole scene is quite beautiful and completes Aragorn's journey to the throne of Gondor.
When the couple approaches the four hobbits, though Frodo in particular, Aragorn cements his greatness by proclaiming that they bow to no one.
Prancing Pony
The four hobbits first set out for the inn of The Prancing Pony, hoping to meet Gandalf there. Instead, they come across Aragorn, called Strider, who rescues them from the chaos at the inn and guides them to Rivendell.
Thus, Aragorn serves as another mentor figure for Frodo and the rest of the hobbits, after Bilbo and Gandalf.
With Arwen, Doubting
While dwelling at Rivendell, Aragorn and Arwen share a conversation where he confesses his fears to her about being Isildur's Heir. Aragorn worries that the same weakness that Isildur had still flowed in his blood, and he is worried that he might not make the right choices when the moment comes.
Arwen assures him that he will succeed, because of the kind of person he is. It's good to see a character like Aragorn confess to fear. Being afraid can keep you alive in the right scenario.
Helm's Deep
During the battle of Helm's Deep, Aragorn comforts a young soldier who is worried about the outcome of the battle. He then takes a large role in directing it.
Considering The Two Towers is the second chapter of the trilogy, Helm's Deep is a battle that is more ordinary than anything else in the trilogy. However, Aragorn's actions, as well as the cause they are all fighting for, make the soldiers' actions brave and worth watching.
Final Battle
During the final battle of the War of the Ring, Aragorn gives encouragement to the soldiers fighting for their cause, quite in the same way that Theoden does.
Though the final battle is full of the great deads of many of the main characters, Aragorn's actions are honorable.
Boromir 
After being shot by the Orcs, Boromir lays dying in the forest, with Aragorn leaning over him. Boromir says that he would have followed him as a king of Gondor, and Aragorn accepts his brother-in-arms' fealty. From a man who earlier proclaimed that "Gondor needs no king", it's a grand gesture.
Therefore, accepting Boromir as he is dying was one of the most heartbreaking scenes in the Lord of the Rings, and makes for a wonderful addition to Aragorn's and Boromir's characters.
Council
During the Council at Rivendell, Aragorn is one of the first men to speak up and say he will journey to Mordor with Frodo to destroy the Ring.
Legolas announces his lineage in order for Boromir and the others to trust him, but it's clear that he is along for the journey early on. Aragorn was the one to escort the four hobbits to Rivendell in the first place, after all.
Eowyn
After they meet, Eowyn is smitten with Aragorn, though he is in love with someone else. He's able to let her down somewhat gently, telling her that what she loves of him is only a shadow and not his real self.
It takes time for her to understand this, though he put it the best way he can, avoiding letting her believe that a romantic future between them was possible.