Elle vs. lui
In French, both elle and lui can relate to the English pronoun "she" or "her", but they are not interchangeable. Their use depends entirely on the grammatical role they play in a sentence: whether she is the one performing an action, or the one indirectly receiving an action.
Elle
A1The word elle is a subject pronoun. It is used when she is the person performing the action of the verb. It's the answer to the question Who?. It is also used as a stressed pronoun, typically after prepositions like avec (with), pour (for), and chez (at the home of).
Elle lit un livre dans le jardin.
(She is reading a book in the garden.)
Elle a préparé un gâteau délicieux.
(She prepared a delicious cake.)
Je vais au cinéma avec elle ce soir.
(I am going to the movies with her tonight.)
Ce bouquet de fleurs est pour elle.
(This bouquet of flowers is for her.)
C'est elle qui a gagné la compétition.
(It is she who won the competition.)
Lui
A2The word lui is an indirect object pronoun. It means to her (or to him) and is used when she is the indirect recipient of the action. It answers the question To whom?. Unlike elle, lui is almost always placed directly before the verb.
Le facteur lui a donné un colis.
(The mailman gave a package to her.)
Je lui parle tous les matins au téléphone.
(I speak to her on the phone every morning.)
Son frère lui a écrit une longue lettre.
(Her brother wrote a long letter to her.)
Tu devrais lui demander la permission.
(You should ask her for permission.)
Ne lui raconte pas cette histoire.
(Do not tell that story to her.)
Summary
The fundamental difference is function. Use elle when she is the subject doing the action (She reads) or after a preposition (with her). Use lui when she is the indirect object, meaning to her, and place it before the verb (I speak to her).