A solar storm occurs when the sun releases more energy than usual, which can be due to an event such as a solar flare.
Solar wind--which is the charged particles emitted from the sun--is mostly deflected by the Earth's magnetic field. The only places on the planet where solar wind reaches our atmosphere is at the poles. This is because at the poles the magnetic field lines are perpendicular to the planet's surface (they become more parallel as you approach the equator), guiding the charged particles into the atmosphere rather than away from it. The result is aurora borealis (sometimes called "Northern Lights")--colorful light emitted by the ionization of atmospheric molecules.
During a solar storm, solar wind is stronger. This can result in more charged particles breaching the Earth's magnetic field and reaching the atmosphere, possibly creating auroras in places other than the poles. Since solar wind ionizes the atmosphere, I don't believe dangerous energy can penetrate all the way to the surface of the planet, so don't worry about the effects of a solar storm on your body.
However, since electronics are more sensitive, there may be power outages and the like. If you think you're in an area that will experience electronic-related issues (the closer you are to the equator, the safer you are), you may want to purchase a surge protector for your important equipment. You should also consider getting a backup power supply for any desktop computers or similar devices you own, because your power may flicker off and on intermittently, which may damage a computer and would certainly make it difficult to use.