A Rat's tail is a quest item used in the Witch's Potion quest. It can be obtained by killing any standard rat after the player has begun Witch's Potion. The item has a 100% chance to drop once you begin the quest.
Brian's Archery Supplies, a store that sells Ranged supplies, has a couple of rats which are very convenient for the player, since the store is located near Hetty the Witch's house.
After completion of the quest, the tail is unobtainable unless the player uses the drop trick to obtain multiple tails during the quest. If the player shows a Rat's Tail to the Wise Old Man after completing the quest, he will tell the player to get rid of it. (Before completing he says "You have brought me a tail of a rat!" and "Yes, I think you should keep it for now.")
For an exhaustive list of all known sources for this item, see here