I've got a few characters who have this icon. I have no idea what it indicates.
The icon indicates a character has a Disease. Most of them give the character 1 or more stat penalties (such as -5 SPD or +100% food consumed), but there a couple diseases which give you a stat bonus as well.
Your explorers can gain a disease from Monster Attacks, Traps, and as a random quirk at the end of a dungeon (non-wiki source]. When you gain a disease from trap or monster you'll see the below animation pop up over their head. (source):
You can see which diseases your explorers by looking in the bottom right section of their status menu. It'll listed all their diseases, and hovering over them will tell you the effects.
There are a few ways to cure a disease. In town you can cure them at the Sanitairum, but that doesn't unlock until you've done a few dungeon runs. A random Internet comment suggests the Sanitarium has a chance to cure two diseases, but I haven't confirmed that myself.
There are a few camp skills that can cure diseases as well. The Plague Doctor can remove diseases using the camp skills Leeches (Target Ally, Heal 15%, Cure Disease, Cure Blight) and The Cure (Target Self, Cure Disease, +20% Disease resistance). The Grave Robber can remove diseases using the camp skill Snuff Box (Target Self/Ally, Cure Disease).
Finally, there's an item called Medicinal Herbs whose description says it cures diseases, but according the wiki and some Internet research it doesn't actually remove it from explorers. According to the Internet you can use them to interact with certain Curio so that whatever negative effect normally has a chance to happen won't, and you'll just get some loot or other bonus. 1 2 3 4