The punch line is the last line of the joke, the line at which the audience understands the point of the joke and laughs. The joke suddenly "hits" them. The audience feels the "impact" of the joke.
It's humiliating to laugh before the punch line, because you are showing that you don't understand the joke at all, and are just laughing to make people think you do.
(The language used to describe comedians in front of an audience is oddly violent. The comedian hopes to "kill 'em dead," and if the audience doesn't react, the comedian "died out there.")
A man in a bar says to the man next to him "I have to ask, from the way you talk, you sound Irish."
The other man says "I am! And where are you from, may I ask?"
The first man says "County Donegal"
The other man says "So am I! Small world, isn't it? I went to St. Columba's, where did you go?"
The first man says "That amazing! I did too. And did Sister Theresa smack you over the knuckles?"
The other man says, "Yes, she did."
The bartender says to another patron, "This is going to be a long night."
The patron says "Why?"
[Up to this point, the joke is not funny. It sounds like it is just a story of an amazing life coincidence. Here comes the punch line:]
The bartender says "It's the Murphy twins--and they're drunk again."