How do you say fat in Japanese?

The Word changes based on what you mean, there is fat as in "fat person" and then a noun used for various fats and oils.

Fats and Oils:

For this case you can use 脂肪 (shibou, fats/grease) or 油 (abura, oil).

Fat Stature:

In this case the Word is デブ (debu). An example is デブになりたくない！ (debu ni naritakunai) which means "I don't want to get fat!".