SP is a stat needed for characters to use their special skills.
In dungeons, hold the A and B buttons until the arrow points to the bar you want to use. There are 4 bars in total, each with their own unique skills. This is one of the stats that cannot be altered in any way except by using certain equipments.
SP can be gained by attacking monsters or equipping an equipment that increases SP regeneration. Certain weapons can even steal SP from monsters, even though monsters do not use SP whatsoever.