"A spoilt brat " or a "little emperor"

Peachy is right we usually use the verb 'to spoil'. Then we would say 'They spoiled their son' or 'their son was spoilt'. You can also use to pamper, to overindulge, to cosset and to mollycoddle ( a lovely word which which means both to be overprotective and indulgent).

Off the top of my head, I'm not sure of the noun but the action is "to spoil (someone)". So someone is described as "spoilt" because they expect everything to go their way.

to describe it means you're looking for an "adjective", not a "noun". But it can also be used as a verb. For example, "Spoiled" (adjective) brat/child ; To spoil (verb) a child. I don't know how to use it as a noun... maybe if you simply call someone as "Spoiled" instead of calling him/her by his real name.. then it becomes a noun. :)

As a variant: pampering/coshering/coddling treatment