Solar flares occur when magnetic energy that has been building up in the upper atmosphere of the sun is suddenly released. This causes radiation from most of the electromagnetic spectrum to shoot towards Earth. Sometimes this flare also allows for a coronal mass ejection (CME) in which the part of the sun's mass, usually contained by the magnetic fields around the star, escapes from the sun and shoot towards Earth.
The most common effect of solar flares is a disruption in radio waves trying to pass through the atmosphere. This translates into communications disruptions with satellites, and well as terrestrial communications over shortwave radio. This is because when the highly charged particles from a solar flare hit the outermost layer of the Earth's atmosphere, the ionosphere, the particles increase the ionization of gases in this region. This leads to a sudden increase in electron production, which causes the interference.
When a solar flare is accompanied by a CME, the result can be the phenomenon known as the aurora. When the highly magnetized CME hits the Earth's atmosphere, what happens depends on the current geomagnetic polarity of the area of the magnetosphere it hits. If the magnetic field around the CME, known as the interplanetary magnetic field (IMF), and the area of the magnetosphere have polarities in the same direction, then the magnetosphere can repel the CME entirely. However, if the IMF around the CME and the magnetosphere have differing polarities, then the IMF around the CME merges with the magnetosphere and briefly connects the magnetic field of the Earth and the sun. The visual result is the light show of the aurora.
If a solar flare is powerful enough, then the effects on the Earth won't be restricted to its atmosphere. When solar flares increase the ionization of the gases in the ionosphere, the buildup of electrons causes a large electric current, called an "electrojet," in the atmosphere. The magnetic field this electrical current creates can in turn induce an electrical field in certain high-latitude regions of the Earth. This electrical field can overload power grids and knock out electricity for these areas.