What is the significance of having the nails brought up to the flame during havdala?
To remind us about the original sin of Adam and Eve. Prior to sinning they were clothed in something similar to nails, after the sin they lost the covering. The remnant on our fingertips reminds us of that. Also, fire was created after that first Shabbat in the garden of eden. After Adam and Eve did teshuva, God gave them fire to light the way. By benefiting from the fire by havdala we are reminded of the power of teshuva.