Conforming to established standards of behavior or manners; correct or decorous.
For example:
Frank was most proper while in the presence of his grandparents until the moment when he was presented with a steak bone and lost all sense of propriety while whirling in a ball of yapping exhilaration...
Now, what you might want to know is that not only does this photo fit definition #1, but also #2, as when Frank looks like this and is also snuggled in, it causes ME to snuggle in and sleep!