Learn Hardcore Portuguese (Brazil): Eu quero o livro.

In Portuguese, nouns are usually accompanied by an article (definite or indefinite). The article o is the masculine definite article, so it specifies that you’re talking about the book. This is similar to using “the” in English to refer to a specific object.

Is Eu quero always necessary, or can I just say Quero o livro?

You can omit Eu and say Quero o livro, and it would still mean the same thing. In Portuguese, the verb conjugation already indicates the subject (eu). Saying Eu quero o livro is perfectly correct but slightly more emphatic because you explicitly state the subject.

Why is it quero and not quer?

Quero is the first-person singular (eu) present-tense form of the verb querer. Quer is the third-person singular form (ele/ela/você). Since you are saying “I want,” you need quero.

Is there a polite or formal way to say "I want the book" in Portuguese?

Brazilian Portuguese doesn’t usually switch to a separate verb form for politeness. However, you can soften the request with Eu gostaria do livro (I would like the book) or Poderia me dar o livro, por favor? (Could you give me the book, please?). These sound more polite but convey a slightly different nuance.