The word "proper" means "correct". It sounds formal, and to most Americans it sounds a little British. It makes me think of men in tuxedos, and women in big white dresses drinking tea.
Of course, "proper" and "properly" aren't always used in such extremely formal situations. You can use "properly" to talk about doing something in the correct way:
He handled it properly and professionally.
Your brain needs oxygen in order to function properly.