Same thing happened to his dad in the Namek saga. Frieza just had to kill his best friend Krillin at the end of a chain of revenge attacks (killing Dende and severely injuring Piccolo) and then go on to threaten Gohan. Bad.Fucking.Idea. Just to prove he's Not Himself, he actually pledges to make Frieza Suffer, and yells at Gohan to go back to the ship.
On that note, Goku had a similar berserk button pressed when Krillin was murdered by Tambourine, and (at least in Budokai Tenkaichi 2's GT story mode, where he isn't incapacitated for the following) when he learns from Android 18 that Android 17 murdered Krillin.
Vegeta gets one of these as well during the Cell saga, when Cell kills Future Trunks. Vegeta - who, up until this point, has either ignored or explicitly rejected any attempts by Trunks to reach out to him - absolutely loses it. Of course, his attack doesn't work, but it's a pretty critical character-defining moment.
Don't harm Vegeta's pride either. It won't go well for you.
Don't try to pimp out Bulma either. Goku learned that the hard way.
Same goes for attacking Bulma in front of him. Ask Beerus how that one felt.
After Vegeta loses to Android 18 he has one of these.
In Dragon Ball Z: Battle of Gods, the first time Vegeta goes at it with Beerus, he's soundly beaten. When he sees Beerus smack his wife Bulma, though, he goes absolutely berserk and actually gives the God of Destruction a run for his money for a short while.
Broly will attack anyone who cries or looks like Goku. This stems from their time as babies in a Saiyan nursery. Goku (Kakarot), in the bed next to Broly, would cry incessantly.
Whatever you do, do not, I repeat, DO NOT wound General Blue's looks in any way, shape or form, or else you'll be extraordinarily lucky if Blue even decides to unfreeze you from his telekinetic ability and stop trying to kill you.
Don't hurt Mr. Satan or Bee when Fat Buu is around.