The sun's magnetic fields are pretty complex and hard to appreciate; however, to make them easier to understand, NASA has built a moving computer model that is aimed at mapping the star's fields.
The model itself shows closed magnetic field lines, and open ones which project out into space. The open lines are colored pink or green while bright spots appear when the closed lines run into each other.
Studying the solar magnetic fields is important because these magnetic explosions can cause solar flares and solar storms. Within the scientific community, many suggest that solar storms could at some point pose a threat to Earth, not just disrupting space travel, but also disrupting our own magnetic fields, something that could destroy our power grids and leave us without electricity for a matter of months.
In fact, such an incident took place in 1859. It was called the Carrington Event, and it resulted in fires and telegraph networks to shut down. Today, however, we rely on electricity a lot more, so such an event would be far more serious.
According to documents released a few months ago, the U.S. Geological Survey actually requested a budget increase to specifically map out geomagnetic fields from the Earth and research ways to protect us in case of a huge solar storm. Hopefully, if it eventually does get funding for that program, and if a massive solar storm does ever occur, we'll know enough to protect ourselves.
Until then, you can check out the psychedelic video below.