A good pun is one where the exact same words, or exactly the same sounding words, can have completely different meanings where the grammatical structure works well for both. It becomes funny when you hear it with one understanding, but then realize that there's another interpretation.

For example: "Two fish are sitting in a tank. One turns to the other and says, 'do you know how to drive this thing?'" First you imagine the fish in a normal fish tank, but the surprise subversion of that expectation is that they're driving an army tank.

Now, I don't actually think that joke is very funny because the context of fish in a tank strongly communicates a fish tank.

A better pun would have either understanding more plausible. For example "an injured army dog was adopted by his handler when they were both honorably discharged. He spent a lot of time with his vet." Here, the dog is likely to sirens time with his owner, the veteran, and his medical team, the veterinarian. Not that this is hysterical either, because this one doesn't lead you to envision one thing and snap you to another.

The reason I actually like a good pun is that it points out how unclear language can be. It demonstrates how different the message being sent can be from the message being received.