Gato vs. felino
While both gato and felino can be translated as cat in English, they are not used interchangeably in Spanish. Gato is the specific, common word for a domestic cat, while felino is a broader, more scientific term for the entire cat family.
Gato
A1This is the standard, everyday word for a domestic cat (Felis catus), the kind people keep as pets. It is the most common and informal term.
Mi gato duerme sobre el teclado de mi computadora.
(My cat sleeps on my computer keyboard.)
La gata de mi vecina tuvo cinco crías ayer.
(My neighbor's female cat had five kittens yesterday.)
¿Le diste de comer al gato esta mañana?
(Did you feed the cat this morning?)
Hay muchos gatos callejeros en este barrio.
(There are many stray cats in this neighborhood.)
Aquí hay gato encerrado, no me fío de su explicación.
(There is something fishy going on here, I don't trust his explanation.)
Felino
B2This is a more formal or biological term that refers to any member of the cat family (Felidae). It can be used as a noun for any cat species (from a house cat to a lion) or as an adjective to describe cat-like qualities.
El tigre es el felino más grande del mundo.
(The tiger is the largest feline in the world.)
La exposición del museo documenta la evolución de los felinos salvajes.
(The museum exhibit documents the evolution of wild felines.)
La ladrona se movía con una agilidad felina por los tejados.
(The thief moved with a feline agility across the rooftops.)
El leopardo de las nieves es un felino muy difícil de observar en la naturaleza.
(The snow leopard is a feline that is very difficult to observe in the wild.)
El veterinario es un especialista en enfermedades de los felinos domésticos.
(The veterinarian is a specialist in domestic feline diseases.)
Summary
In essence, use gato when you are talking about a house cat in normal conversation. Use felino when you are speaking in a more scientific or formal context, referring to the entire cat family (including lions, tigers, etc.), or describing a cat-like quality.