Learn French: Les parents gâtent la fille et lui donnent un doudou. - The parents spoil the girl and give her a cuddly toy.

As an indirect object pronoun, lui means both 'to him' and 'to her'. The verb is donner à (to give to), so it takes an indirect object. Therefore, lui donnent means 'give to her'. If the verb took a direct object, you would use la.

Is un doudou a normal word for a toy?

Un doudou is a specific, everyday word for a child's favorite cuddly toy or security blanket. For a general stuffed animal, you would say une peluche, but doudou is extremely common in French for a child's comfort object.

Why is there no subject before donnent?

Because the subject is still Les parents, you don't need to repeat it after et (and). However, notice that the pronoun lui must still be placed immediately before the verb it belongs to (donnent).

What is the infinitive of gâtent?

The infinitive is gâter (to spoil). It is a perfectly regular -er verb. The circumflex accent over the 'a' doesn't change how it is conjugated, but it is an important part of the spelling!