In the world of Avatar, the bending arts have multiple sources. Each Nation is associated with a particular Bending element; why that association is present has yet to be fully clarified. From observation, however, it appears to have both a genetic element and a personality element.
We see that children always have the element of their parents; when the parents have a different Bending element, the children can inherit theirs from the mother or father, like with Aang and Katara's children, or with Mako and Bolin. This would seem to indicate a genetic predisposition towards Bending a particular element.
At the same time, we see throughout both Legend of Korra and The Last Airbender that Bending draws its power from your personality, or Chi. Airbenders focus on evasion and speed. Waterbenders use adaptability and versatility to power their Bending. Earthbenders use stubbornness and resistance. Firebenders use energy and aggression. This can even be seen in the behaviors of the four Nations on a large scale. The Earth Nation focused on defense, and was hesitant to go on an offensive campaign versus the Fire Nation during the 100 Year War, while the Fire Nation's aggression and energy kept them going strong for the whole conflict.
This is where the idea of jings fits in. As discussed by Aang and Bumi, positive jing is when you are attacking, negative jing is when you are retreating, and neutral jing is when you are waiting and listening. Airbending focuses on negative jing. Firebending focuses on positive jing. Earthbending focuses on neutral jing, and Waterbending focuses on alternating between positive and negative as the need arises.
Aang had a hard time with Earthbending because it was against his personality. He avoided combat at all times, if possible. He had trouble exerting his will against the environment around him, preferring to practice avoidance and mitigation. Waterbending was natural for him, as he was very adaptable to his surroundings. Korra, unlike Aang, does not have an Airbender's personality. She is strong willed, and aggressive. Her quick mastery of Firebending is because she can focus her aggression and inner energy into it. Airbending, however, requires peace and evasion. It's simply not in Korra's personality to avoid conflict.
Korra can Firebend easily because she has a Firebender's heart. Her understanding of positive jing allows her to easily focus her aggression. She has trouble with Airbending because her inner Chi does not line up naturally with what is necessary to Airbend. Her negative jing is too weak. It is the Avatar's role to maintain balance within the universe. That is why it is vital for them to master the four elements. Only when they have fully brought balance to their Chi can they hope to bring balance to the world around them.