Character Analysis
We all know who Darth Vader is. A cultural icon, this world famous villain has been in novels, video games, and comic books. He's been a toy, re-imagined as pop art, and even had his mug slapped on a cereal box. The three prequel Star Wars movies are dedicated to telling how Anakin Skywalker became the most feared cyborg in the galaxy. Darth even has his footprints immortalized on the Hollywood walk of fame.
However, our purpose here is to analysis Darth Vader as he appeared in the original Star Wars film. We're focusing on this character as presented in his original film appearance. Since we don't learn about his relation to Luke until The Empire Strikes Back, we won't be dissecting those daddy issues. Likewise, although we know that he and Obi-Wan have history, we'll only cover the history as detailed in this film, meaning the motives elaborated don't count here.
Original Villain
Darth Vader is a much simpler character in the original film. Despite his later growth in Star Wars mythos, in the first film he's a cold-hearted villain.
Consider his first appearance. Before we even know his name, we see a man dressed in all black entering the aftermath of the battle aboard the rebel cruiser. His faced is covered with an expressionless mask, and his mechanical breathing can be heard in the silence. He looks briefly at the dead bodies littering his path before stepping over them and entering the ship.
We know that guy isn't entirely human, and everything about the scene tells us this. His mechanical breathing and the control panel on his chest let us know he's some sort of man/machine hybrid, and the way he nonchalantly observes the death and destruction around him tell us he's a sociopath.
Later, Obi-Wan will tell Luke,
"A young Jedi named Darth Vader, who was a pupil of mine until he turned to evil, helped the Empire hunt down and destroy the Jedi Knight. He betrayed and murdered your father."
We've already figured out what kind of character he is well before this, though. This line just provides some history for why he is evil and gives us a red herring for the reveal in the sequel.
For the rest of the film, Vader acts like the Empire's goon:
LEIA: Darth Vader, only you could be so bold. The Imperial Senate will not sit still for this. When they hear you've attacked a diplomatic—
VADER: Don't act so surprised, your highness. You weren't on any mercy mission this time. Several transmissions were beamed to this ship by rebel spies. I want to know what happened to the plans they sent you.
The implication of Leia's line is that Vader stepped outside the law when he attacked and boarded her ship. Vader's own speech lets us know that he doesn't care. His goal is to get the plans by any means, and those means include, but are not limited to, murdering people, torturing Leia, and even Force-choking a fellow Imperial Officer for getting sassy with him.
Finally, when the Rebels begin their offensive on the Death Star, Darth Vader takes the fight to them, personally shooting down more Rebels than any other fighter.
He's just a jerk.
Black Knight Crossing
Vader is a science fiction reboot of a character type called the Black Knight. The black knight is a classic character that has its origin in chivalric romances, such as various characters from the Arthurian legends, perhaps the most famous of whom is a big baddie named Mordred.
What is a black knight, exactly? He's a) dressed all in black, to show how evil he is b) totally powerful and in control c) always carries his weapon and d) acts as the central villain in the hero's tale.
Sound familiar? Darth Vader is the archetypal Black Knight put in a rocket and launched into space. He checks every box on that list, and in some ways, ups the ante—his sword is, after all, a laser sword.
Usually, these Black Knight characters are master-less, too, having dishonored themselves for unchivalrous conduct before being kicked out of their order.
Although we don't get much about Vader's history in this film, Obi-Wan confirms that this is true about Vader when he tells Luke about Darth turning to the Dark Side. Also, a short exchange between Obi-Wan and Vader further confirms Vader's disgrace from the Jedi order:
DARTH VADER: I've been waiting for you, Obi-Wan. We meet again at last. The circle is now complete. When I left you, I was but the learner. Now I am the master.
BEN: Only a master of evil, Darth.
VADER: Your powers are weak, old man.
Ol' Vader clearly views himself as having supplanted Obi-Wan as the master. However, unlike Obi-Wan, he has no student to teach. While Vader was once a part of a group, the Jedi, he has now become a lone wolf. Obi-Wan's response that he is "[o]nly a master of evil" points to his moral alignment but also suggests that Vader's mastery is meaningless because it brought him suffering, disgrace, and—maybe most importantly—no companionship.
The information provided does what is necessary for us to enjoy Darth Vader in this film: he's a villain. We want to see him defeated at the end, and we get just that. After almost killing Luke in the Death Star trench, Vader's TIE fighter is launched into space. Defeated, Vader flies into space, remaining the mysterious figure of a true Black Knight.