On the planet Krypton, the House of El is well known in science and in discoveries. Jor-El and Zor-El were chief among these scientists, and among them, Zor-El eventually fell for and married a kryptonian woman named Alura. They even had a child well before Jor-El and his wife. The girl was named Kara and raised among the best schools and training in Krypton. She lived a fairly standard life, that is, until the discovery of Krypton's impending destruction by the house of El.
Once it was discovered Krypton was a ticking time bomb, the house of El sought to save their youngest. Jor-El created a ship for his infant son, while Zor-El created one for his teenage daughter. The plan was that both ships would be launched at roughly the same time and sent to Earth, where they would possess great powers and be a great treasure to their adopted planet. To that end, Kara's ship was modified to teach her about Earth Society, put her in stasis, and prepare her body with yellow solar energy so she could protect her infant cousin from the people of Earth as necessary, should they be hostile.
Unfortunately, at the time of escape, Kara's ship was delayed momentarily. She was sent out, but only moments before the planet exploded, and shortly after her cousin. Her ship was caught in the wave from the destruction of Planet Krypton and damaged, slowing it considerably as well as causing a problem in the social understanding systems. She was still receiving the training, but missing some key elements. The ship continued toward Earth, but at a greatly reduced speed, such that her cousin would arrive and live to be an adult as a 'human' before she would ever reach the Milky Way galaxy. Her ship crashed into the ocean amidst a shower of debris and meteors, and Kara escaped into the world.
After some time in chaos, and some time in study, Kara Zor-El would become Supergirl, much as her cousin has taken on a similar heroic name.
Kara Zor-El is a dedicated hero. She won't stop when she says she is going to do something, and she will certainly go through any attempts to ensure that she is true to her word. That means, if she tells you she will help you, you are best to believe she will do everything in her power to follow through on her promise.
Kara Zor-El defiantly has a bit of loneliness to her. She may have a cousin, but she was supposed to take care of him, not the other way around. She is the last 'daughter' of Krypton, and being the last adult survivor of her race that grew up within that society has left her with a deep loneliness.
Kara is very much a hero, it runs in the El Family blood. She regularly puts herself in great danger to help people, and including doing so even when there is no reward or even little knowledge that she was involved. Kara doesn't do it for the glory or the fame, through and through, she does it because she believes it is right.
Being the last adult survivor of Krypton has left Kara with a deep level of survivor's guilt. She thought her parents should have survived, more of her people, anyone. She carries a deep amount of guilt for surviving, but she uses that to push herself onward and give the kryptonians an even greater heritage.
Kara is defiantly the new girl on Earth, and it shows in her actions. She's not good at social interactions, and is regularly one to make inappropriate social comments or graces, due to the fact that well, it's just beyond her grasp. Just like the new girl in town, only to a much higher degree.
Kara's eyes are incredibly different from human ones, and she is capable of seeing through any material on earth, save Lead, with her X-Ray vision. She is also capable of releasing concentrated solar energy in the form of 'heat vision', which operates as hot as a plasma cutter/welder. She is also capable of telescopic vision, much like that of an eagle, where she can see things at great distances. If she truly focuses, she can even see other spectrums of light and energy, though it strains her to do such.
Kara's muscles are super dense, and she is able to easily lift over 100 Tons of weight and repeatedly. She can punch through concrete bunkers and heavily reinforced buildings.
Kara's skin is super dense, and she is able to resist bullets, explosives, high caliber weapons fire and small melee weapons equally. Essentially, there is no weapon on earth created by science that can actually hurt her or pierce her skin, though high end explosives and concussive weapons will still cause her distress.
Kara has superhumanly enhanced senses, including taste, smell, and hearing. She can smell things at great distances; though not distinguish them as a dog might. An example would be her smelling a fresh cup of coffee from a mile out. She can also hear subtle whispers from miles out, if she hones in and listens, much as she can with her sense of smell.
On the ground, Kara is superhumanly fast, and can run at speeds breaking that of the Land Speed record (763 MPH), and in the air, is just as fast if not faster, as Kara is able to fly at speeds even higher than that, though doing so tends to cause problems with the weather and create massive sonic booms.
Kara can fly, quite simply. She defies the laws of gravity and can fly through the air, even up to and through outer space, but she needs to remain somewhere relatively close to an oxygen lair to breath, rather than just flying through deep space.
Kara's lungs are changed tremendously from those of humans, and she possesses the ability to breath in incredible amounts, taking in things such as air-based toxins and poisons to clear them quickly, or she can expel breath at great speeds to blow away such toxins and poisons with great force, such as some of the strongest winds of hurricanes and tornados. She is also capable of modifying the temperature of this breath, bringing it to below freezing to actually freeze water molecules in the air and freeze someone in place.
Kara has an alien physiology, due to the fact she is an alien. Essentially her organs and body operate on a different level from those of normal humans, such as processing yellow solar radiation to power her cells. However, it makes her body a marvel for doctors and difficult for attackers to hit a 'nerve' spot. This also makes her immune to almost every toxin and biological that affects humans, given the fact her physiology is different.
Kara is not necessarily a trained fighter, but she is skilled at brawling and fighting. She can fight as necessary, and utilize her powers to assist her 'fighting style' which is basically the same as any 'street fighter' or generally untrained individual.
Kara lived in Kryptonian society. She was a 'teenage girl' in it, and by most terms, that means adult. She's fluent in Kryptonian legends and science, as well as the technology. Essentially, what a human adult would know of their society and its intricacies, Kara understands for Krypton.
Kara is fluent in her home language of Kryptonian, however, given her studies and the system, she is very quick to pick up and understand other languages. For instance, a bit of television, and some time with her ship, and she might be fluent in English, or Spanish as necessary.
Kara Zor-El has access to a significant amount of krypton-based technology in her ship. It has built in stasis, health restoration, and yellow solar batteries. Essentially, the ship is keyed in to her DNA, so she is the only one who can use it. Unfortunately, the flight engines are blown so it cannot fly, but the rest can be used as necessary, including the Kryptonian knowledge stored within.
Kara has access to the Fortress of Solitude, and even lives there partially to have a base of operations as Supergirl.
Simply put, Kara has no special abilities against magic or magically enchanted weapons. A magical weapon will hurt her just the same as a normal weapon would hurt a normal person. Magical attacks are just as efficient against Kara also, so Magic is a tricky situation.
Remnants of Kara's home planet have become radioactive and provide bad effects to Kara. Red is unpredictable, green strips her powers and makes her physically ill and weak, and there are other varieties yet to be discovered.
No one on Earth can know that Kara Zor-El, Supergirl, is actually a relative nobody blonde that is cousin to Clark Kent. After all, that kinda makes it obvious who Superman is, huh?
Kara is an alien. She may know about earth society, but she doesn't understand it. Society is a difficult thing even for a person, so she has incredible trouble interacting in a secret identity method. Add to that the fact she really can't be checked up by a doctor due to her alien physiology, and it's very apparent her alien heritage is both a blessing and a curse.
Kara has an unusually healthy appetite. She eats as much as she possibly can to help maintain her body and help power the solar energy. This is more of a social flaw though, as she often stands out like a sore thumb, especially when it comes to her eating in a mall.
Unlike her cousin, Kara is fairly new to possessing her powers. While she doesn't activate them without meaning to, she certainly has issues controlling the usage of them. Heatvision comes out hotter than intended, she doesn't pull her punches when attacking with super strength, and often goes unnecessarily fast under super speed. The end result is that she tends to burn her solar power faster, and can cause unnecessary property damage due to results of her powers.
Simply, all of Kara's powers are fueled by her ability to absorb and process solar energy. If she were to operate outside of an area where she can receive yellow solar energy, she will begin to burn through her reserves, while a red sun will empty her cells and leave her powerless. In either essence, in the absence of a yellow sun, Kara will have no powers.