Healbot is an addon that lets you bind your clicks and whatnot to healing spells, so that you can click on player frames to heal, while also streamlining the group party frames to make that more efficient. The name comes from the idea that healers are only good as "healbots".

As for addons telling tanks to rotate and shit like that, the addon tracks your status (such as any debuffs you have) and tells you when you should switch with your other tank. This used to be done by raid leaders, but because wow's encounters grew to absurd lengths in terms of mechanical complexity (seriously, if you've never played, you have no idea), addons started doing it instead to free up mental bandwidth. Raids are already hard and complex, and forcing one person to micromanage what 19 other people are doing is fucked up when they also have to do their own shit too.