Learn Portuguese (Portugal): Tu tens um caderno aqui.

In Portuguese, the word for 'a' or 'an' changes depending on the gender of the noun. Caderno (notebook) is a masculine word (it usually helps that it ends in -o), so we must use the masculine article um. If the word were feminine, like caneta (pen), you would use uma.

Where does the word tens come from?

Tens is the 'you' (informal) form of the verb ter (to have). Ter is one of the most common irregular verbs in Portuguese. Whenever you use tu (you), the matching form is always tens.

Can I leave out the word Tu and just say Tens um caderno aqui?

Yes, absolutely! Because the verb form tens is only ever used with tu, the listener already knows who you are talking about. Native speakers often drop the pronouns to sound more natural.

How would this sentence change if I were speaking to someone formally?

To be formal, you wouldn't use tu. You would use você, which requires a different verb form: tem. The formal version would be Você tem um caderno aqui or simply Tem um caderno aqui.