I think this is one of the funniest expressions that Jane Austen uses. Although I understand what she means, I can't quite arrive at a definition that suits me for what was meant by a 'puppy' in that time.

We read that Elinor noticed Robert Ferrars's "impertinent examination of their features, and...the puppyism of his manner in deciding on all the different horrors of the different toothpick-cases presented to his inspection"

Mr. Knightly thinks Frank Churchill is a puppy:

"[My idea of Frank Churchill] is...he will be the most insufferable fellow breathing! What! at three-and-twenty to be the king of his company -- the great man -- the practised politician, who is to read every body's character, and make every body's talents conduce to the display of his own superiority; to be dispensing his flatteries around, that he may make all appear like fools compared with himself! My dear Emma, your own good sense could not endure such a puppy when it came to the point."

Yet Mrs. Elton thinks he is not

I am happy to say that I am extremely pleased with him. You may believe me. I never compliment. I think him a very handsome young man, and his manners are precisely what I like and approve -- so truly the gentleman, without the least conceit or puppyism. You must know I have a vast dislike to puppies -- quite a horror of them.