The pendant came into being after creating a magic stone while constructing the barrier around Abigail's sealed power. After creating the pendant, it was passed down through the Lowell family line. Eventually, it fell into the hands of Nina Lowell when her father passed away.
Alan's Tear is a magical stone with power great enough to ward off lesser demons, and in some cases capable of destroying them. However, the necklace doesn't react to half-human hybrids, Dante is able to hold it and be in full contact without suffering any injury or negative effects. Demons who possess little, to no power at all aren't as greatly affected by the stone's warding, but are still open to some degree of reaction with Alan's Tear. Sid, a lesser demon attempts to touch the stone, his lack of demonic power results in his hand being burned badly, despite this he comes away from the encounter alive and practically unscathed.
Alan's Tear's true purpose is being able to seal away the power of Abigail. Undoing the process requires a ritual with various demonic items arranged into the seal in order to penetrate it's barrier. Lady once stated the barrier was so powerful, not even the smallest ant could pass over its boundaries.