Healing Potions will immediately heal a certain amount of health (red bar), different from Food or Mushrooms.
The Basic Healing Potion will heal health 250 points at once. It can be obtained from Corrupted Creatures as a loot or pet-harvest, or crafted with any type of Mushrooms and Flowers from a starting crafting recipe.
The Advanced Healing Potion will heal 600 health points at once. It can be crafted after having crafted or obtained Basic Health Potions from Mushrooms of any kind, Leafi Leaves of any kind and BossHog Tusks.
Healing Potions won't be used up when no healing is needed, but they also have a short cooldown, so you will have to wait a little until you can gulp down another potion of the same type.
The same goes for Mushrooms that only heal a little health over time but also have a cooldown. However each potion and type of Mushroom has their own cooldowns that won't hinder you to consume other types right away.
The Health Regeneration Potion will heal over time for 10 minutes, significantly faster than Healing Beacons and/or Mineral Water will. The effect can seemingly be stacked - meaning that healing over time is accelerated a little when using a Regeneration potion plus being next to / in / on top of a Healing Beacon and/or unit of Mineral Water.
Health Regeneration Potions can slow down the poisoning effect of Corruption, but not negate it (they cannot make it stop). The same goes for all Mushrooms, as they also heal over time at about the same speed as Regeneration Potions for 30 seconds each, but only for a small amount (1/10 or less of your health-bar) plus with a cooldown for every mushroom-type seperately.
"Normal" health-bar means: without any buffs (+100 or +200 HP can temporarily be added by consuming special types of Food).
The "Medium Potion" is not in the game any more right now (as of R23 in October 2015). It might still be bestowed as a rare gift now and then, like as a content of item packs granted by Promo Codes.