[Spoilers main] How surnnames system works in Westeros?

If a bride was taken from different house then she change her surname, right? Like Catelyn Stark or Lisa Arryn (both nee Tally) but why Cersei Lannister or Aemma Arryn kept their surnames? Or it doesn’t depends on their will? I've heard it may has smth to do with the house greatness then how to define which house is moe great (for example, in Catelyn and Lisa's situations)

For me it seems like if you have a powerful house then you will still be called by your previous surname, but how to find out who is more noble or less... I am lost...