Chat vs. félin
In French, while both chat and félin can refer to cats, they are not interchangeable. Chat is the common, everyday word for a domestic cat, whereas félin is a broader, more scientific term for the entire cat family or an adjective meaning cat-like.
Chat
A1This is the most common and standard word for a domestic cat. It's the word you use to talk about your pet, a stray in the street, or house cats in general. It is the direct equivalent of the English word cat.
Mon chat dort sur le canapé toute la journée.
(My cat sleeps on the sofa all day long.)
Le chat de la voisine est très affectueux.
(The neighbor's cat is very affectionate.)
Il y a beaucoup de chats errants dans ce quartier.
(There are a lot of stray cats in this neighborhood.)
J'ai toujours voulu avoir un chat depuis que je suis enfant.
(I have always wanted to have a cat since I was a child.)
Il ne faut pas réveiller le chat qui dort.
(You shouldn't wake a sleeping cat (equivalent to let sleeping dogs lie).)
Félin
B2This is a more formal or scientific term. As a noun (un félin), it refers to any member of the biological cat family (Felidae), which includes lions, tigers, cheetahs, and panthers, as well as domestic cats. As an adjective (félin or féline), it is used to describe a quality as being cat-like, such as grace, agility, or appearance.
Le lion est un grand félin qui vit en Afrique.
(The lion is a large feline that lives in Africa.)
Cette exposition au musée présente les différents félins du monde.
(This museum exhibit presents the different felines of the world.)
La gymnaste se déplace avec une agilité féline.
(The gymnast moves with a feline agility.)
Elle a un regard félin, à la fois perçant et mystérieux.
(She has a cat-like gaze, both piercing and mysterious.)
Tous les chats sont des félins, mais tous les félins ne sont pas des chats.
(All cats are felines, but not all felines are cats.)
Summary
In summary, use chat for the everyday domestic cat. Use félin when referring to the broader cat family (including wild cats like tigers and lions) or as an adjective to describe something as cat-like. You would call your pet mon chat, but you would describe its graceful jump as a saut félin.