Some tattoo shops exist primarily on covering up names of past lovers, though a surprising number of tattoo artists will tell you that they've added NEW names to people who've had old ones covered up.
No matter the strength of the relationship, do not get your partner's name tattooed on you. It's a bad idea.
It is generally safe to get your parents' or childrens' names tattooed - those presumably aren't changing anytime soon. If you are male, it's a good idea to have a paternity test done on the child before getting the tattoo.