The noun il complimento sometimes means the same thing as in English: the compliment. It's used a bit differently, and is often synonymous with "congratulations."
When you want to say, "Nice job!" you might say, Complimenti!
Complimenti, mamma, ma qual'è il tuo segreto?
Very nice, Mom, but what's your secret?
Caption 33, Adriano La granita al limonePlay Caption
But there is another way complimenti is used, and it's important, especially if someone invites you to their home and you are not sure how to act. In order to put you at ease, they might say, non fare complimenti. It means, "Relax, you don't have to be formal." This is especially true at the dinner table. The host or hostess might say, Serviti, non fare complimenti. So you can go ahead and take seconds...
Sì, però, è che non vorrei... -E non fare complimenti, scusa.
Yes, but it's that I wouldn't want... -And don't say no out of politeness, sorry.Play Caption
Grazie, eh. -No, prego, non fare complimenti, ah.
Thanks, huh. -No, you're welcome, don't stand on ceremony, huh.Play Caption