The below is a list of what Badass could be depending on the person and the situation.
Badass as "Cool, awesome"The word "badass" is quite often used as yet another synonym to "cool, awesome, sweet, rad, etc". The usage can range from contexts overlapping with other uses listed on this page ("that stunt was badass", "this is a badass fight") to absolutely anything ("what a badass guitar solo", "that's a pretty badass car", "bro, your gran's embroidery looks fucking badass, man"). This is Not a Trope in itself, but we do maintain specific sections to list awesome characters and moments in fiction: see Gushing About Characters You Like and Moment of Awesome.
Badass as FighterAny competent warrior; from the best fighter in the universe to just any trained fighter. If fighting is required, it's best if you can handle it well enough to avoid embarrassing yourself. A Badass is a character that is known for their ability to fight and carry their own weight. If Alice can defeat three grown men attacking her at once, that's pretty impressive. However, she might be outclassed by the Badass characters who can defeat three hundred grown men at once.
Badass as Escapist CharacterAn idealization of the author or the audience. These characters are exceptionally brave, strong, skilled etc. Some might accuse them of being too perfect or powerful. A Marty Stu is basically an extreme badass. Quoting from the page, "He's an unstoppable fighter, a rogue agent, a fearless freedom fighter, a master of disguise." Also, quoted from the page, "devastatingly handsome (or if not, possessed of a strange, saturnine magnetism) and desired by all significant women".
Badass as Tough and Resilient"The character stays strong despite pain and hazards due to strong willpower". This is where the Badass Pacifist comes from. They don't fight back but their opponent can't hurt them anyways. Determinator covers those characters who are renowned for their limitless willpower; Made of Iron for characters who can't be slowed down by physical injury and don't have an ability like Super Toughness.
Badass as Rule of CoolThis was the definition we had for a while. This is a character who does improbable stunts because it is cool.
Typically, Badass will come down to presentation. Often, even supposedly "normal" badasses can pull off stunts that would be considered highly unlikely in real life, such as fighting off a dozen soldiers at once or shrugging off what should really be a fatal or incapacitating injury. However, many badasses are not limited to what would be passingly reasonable in real life, allowing all sorts of superpowers to come into the equation as necessary, or perhaps their appearance or role is drastically at odds with their ability, such as a little girl tossing adult men like rag dolls.