There are six types of pests, each hindering your work in a different way.
Bugs are brought by your guests, who leave them on beds and chair. Bugs destroy furniture. You need to either order one of your workers to clean the given object or install a bug repellent.
Boars dig in your garden, destroying crops and objects around the inn.
Foxes kill chickens.
Liquerats (small, rodent-like creatures) steal alcohol.
Each pest can be dealt with by using/employing a particular pet (cat or dog), workers or objects. Cats catch rats and liquerats. Dogs chase away pests around the inn (though one may not be enough for a daer). They require a dog house and a bowl.