My dog's got no nose...

My dog's got no nose.
How does he smell, then?
Awful.

This joke is a bit of a pun, but in this case, the pun appears to be based in a linguistic misunderstanding instead of intentionally crafted. The joke is that person A (the person who said "My dog's got no nose.") misunderstood the query that was a response to his original statement as a bit of a non-sequitur. The question is understood (to the listener of the joke) to be asking how the dog is capable of detecting smells, but person A understood it to be asking what the dog smelled like.

Both are reasonable interpretations of the sentence, but given the setup, the second interpretation seems more likely.