Learn Polish: On woli nie jeść tłuszczu. - He prefers not to eat fat.

Just like in English, you can follow woleć (to prefer) directly with the infinitive (dictionary form) of another verb. Here, the main verb woli is followed by the infinitive jeść (to eat).

Why is nie placed before jeść instead of before woli?

It depends on what you want to negate. He does have a preference, so woli stays positive. His preference is 'not to eat', so we put nie directly before jeść. If you said On nie woli jeść, it would mean 'He doesn't prefer to eat', which is a completely different meaning.

Why does the word for fat change to tłuszczu here?

This is a great example of the Genitive of negation. Normally, the verb jeść takes the Accusative case (which would just be tłuszcz). However, Polish has a rule where negating a verb changes its direct object to the Genitive case. Since the verb is negated (nie jeść), the object becomes tłuszczu.

Why does the Genitive form of tłuszcz end in -u instead of -a?

Tłuszcz is a masculine inanimate noun. In the Genitive case, these nouns end in either -a or -u. While there is no perfect rule, uncountable nouns like substances and materials usually take the -u ending.