"Art for the Sake of Art"
This phrase, quite conveniently, uses the same word order in both English and Latin.
Ars, artis (artium) is a third-declension feminine noun. It can mean "art" in the sense of paintings and sculptures, but can also be more abstract, like the "art" of writing (i.e. the skill and experience required to be a good writer). Ars is the nominative "art", artis is the genitive "of art".
Ex: Ars Amātōria, "The Art of Love", a poem by Ovid.
Grātiā is one of the very very few prepositions taking the genitive case, meaning "for the sake of". It started out as the ablative singular of the noun grātia (with short a), which originally meant "sake". But just like in English, the noun became uncommon except in the fixed expression "for the sake of..."; grātia came to mean "thanks" instead, and that's the meaning it has in the Romance languages (Italian grazie, Spanish gracias, etc). So grātiā was re-analyzed as a preposition.
Ex: exemplī grātiā (e.g.), "for the sake of example".
So ars is nominative singular "art", grātiā is a preposition "for the sake", and artis is genitive singular "of art".