Kara Danvers, aka Supergirl, isn't as famous as her cousin Kal-El, aka Clark Kent, aka Superman, but that doesn't mean she's a B-list superhero. After all, Kara has the same basic set of powers as Superman does and the only significant difference between them is that he's had a longer period of time to practice with them.
Even so, despite the serious competition, Kara still managed to become a certified superhero of her own and has defeated many villains and enemies and saved countless lives in the process. She has many friends between her fellow superheroes, but there are also those who despise her.
Despises: Batman
It's not entirely accurate to say that Batman despises Supergirl... but he certainly doesn't trust her, at least not at first, and the feeling is perfectly mutual. No matter how hard Superman tries to bring these two together, they just don't seem to be able to deal with each other in a civil way. Batman is naturally suspicious of any heroes he doesn't know and he's not convinced that Supergirl can handle her powers without presenting a threat to herself and to everyone around her. They eventually reconcile but the start of their relationship is rocky, to say the least.
Respects: Damian Wayne
Damian is the only son of Bruce Wayne and Talia al Ghul and certainly gets a lot from his parents. Damian is often snarky to others, doesn't trust their abilities, talks back, and is generally difficult to get along with, especially if you don't know him well. But he's also immensely loyal to his allies and the people whom he respects, albeit oftentimes grudgingly. That's the case for Supergirl as well. Damian and Supergirl have teamed up in the past on several occasions and even though Damian would never admit it aloud, he respects Supergirl and her fighting skills.
Despises: Green Arrow
There's a big difference in character between the Green Arrow from the comics and the one on the TV show, Arrow. The Oliver Queen played by Stephen Amell is much more serious, and is in many ways more like Batman than the comics' Green Arrow. That also explains his initial attitude towards Supergirl... which is far from friendly, to put it mildly. Oliver doesn't trust Supergirl at first since she's an alien with extraordinary powers that completely change the way Oliver looks at the world. But he completely turns around later on and even sacrifices his own life to save Kara's.
Respects: Mon-El
Many superheroes have a special talent for agonizing other heroes against themselves, at least at first, but then they later become friends with their former enemies. Such was the case for Kara and Mon-El, another alien survivor who ended up on Earth.
Supergirl and Mon-El shared many adventures together and stood side by side, even though their relationship also had its fair share of disappointments, awkward moments, and other problems. Mon-El might not be as well-known as other DC heroes, Supergirl herself included, but he still played a significant part in Supergirl's life, so he deserves a spot on this list.
Despises: Powerboy
Even superheroes are only people (even if they come from space) and as such, they also make everyday mistakes, sometimes slight, sometimes more severe. Their romantic relationships are no exception to this rule. Kara had several significant crushes both in the comics and in the TV show Supergirl. One of her comic book boyfriends was Powerboy, who unfortunately wasn't as super as Supergirl deserved. Their relationship started out okay but Powerboy eventually started to mistreat Supergirl. Their romance became abusive and they broke up later on. So it's guaranteed that Supergirl and Powerboy will never become a power couple like Superman and Lois Lane.
Respects: Batwoman
Similarly to Kara, Kate Kane might not be as well-known as another member of the Bat-Family, but that doesn't make her any less impressive. Kate is a capable hand-to-hand fighter and she takes on the mantle of Batwoman in the comics, which means she gets to work with other superheroes, Batman himself included. As for the relationship between Kara and Kate, if you have seen the recent Arrowverse crossover Crisis on the Infinite Earths you must have noticed that these two superheroines work well together and they also get along well in their civil life. So when Batwoman decides to come out as a lesbian in the press, it's Kara who writes the article about her.
Despises: Rose Wilson
Rose Wilson might be one of the members of Teen Titans but that doesn't make her absolutely heroic. To be fair, it's not entirely Rose's fault. The family you're born in can affect you a lot, and Rose's father, as her surname suggests, is no-one else than Slade Wilson, one of the most dangerous fighters and villains of the DC superhero universe.
He's also an excellent marksman, and Rose certainly inherited this talent from him, as she once shot Supergirl and Wonder Girl while the two heroes were hugging. If you want to know more about Rose, you can read the comics or check out the TV show Titans where she also appeared.
Respects: Wonder Woman
If you know Wonder Woman, it doesn't really come as a surprise to you that she's kind-hearted and generous to her friends, allies, and she's also merciful even towards her enemies... unless they force her hand. Wonder Woman quickly takes Kara under her protective wing as soon as she meets her, as she knows what it's like to come into a strange world you've never been to before where you have to build a brand new life. Wonder Woman helps with Kara's training, brings her to Themyscira where Kara finds new friends and continues to help Supergirl in the future as well.
Despises: Cyborg Superman
Cyborg Superman is one of the complicated DC characters who cross the line between hero and villain. He sometimes acts in a heroic way and then steps to the side of villains. For example, he helped the Suicide Squad defeat mad general Zod in the comics, but he originally planned to work alongside Zod, until he realized that a bigger threat was looming on the horizon and the only way to defeat it was to turn against Zod and work with the Suicide Squad instead. Cyborg Superman may have been partially inspired by Superman, but he certainly isn't as heroic as him, and he and Supergirl have clashed in the comics.
Respects: Superman
Honestly, did you expect that the Man of Steel himself wouldn't make an appearance on this list? After all, he and Kara are family, and two of the last Kryptonians alive. When Kara first arrives on Earth, Superman is overjoyed at finally finding another (non-hostile) member of his species. These two quickly become close and are great friends who respect each other's personalities and skills. Superman can be overprotective of Kara but he means well as he views her as his little sister or even maybe a daughter-like figure, considering their age difference. He's also the one who finally manages to persuade Batman to give Kara a chance.