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). There are a few camp skills that can cure diseases as well.

A random Internet comment suggests the Sanitarium has a chance to cure two diseases, but I haven't confirmed that myself. In town you can cure them at the Sanitairum, but that doesn't unlock until you've done a few dungeon runs. It'll listed all their diseases, and hovering over them will tell you the effects. You can see which diseases your explorers by looking in the bottom right section of their status menu. When you gain a disease from trap or monster you'll see the below animation pop up over their head.

Your explorers can gain a disease from Monster Attacks, Traps, and as a random quirk at the end of a dungeon ( non-wiki source]. 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. The icon indicates a character has a Disease.

Remove affliction darkest dungeon