Learn Portuguese (Portugal): Eu compro o pão no supermercado.

To make the 'I' (eu) form of a regular '-ar' verb like comprar (to buy), you chop off the '-ar' at the end and replace it with '-o'. That gives you compro (I buy).

Is the word Eu strictly necessary here?

Not at all! Because the '-o' ending on compro already makes it clear that 'I' am doing the buying, you can drop the pronoun and just say: Compro o pão no supermercado. Keeping Eu just adds a little emphasis.

The translation says 'in the', but I only see the word no. How does that work?

In Portuguese, the word for 'in' is em, and the masculine word for 'the' is o (matching the masculine noun supermercado). When em and o appear together, they always merge into the single word no ('in the').

Could I just say Eu compro pão without the o?

Yes, you can! Eu compro pão simply means 'I buy bread'. However, Portuguese speakers use definite articles (like o, meaning 'the') much more freely than English speakers do. Including it here is completely natural, even if you are talking about bread in general rather than one specific loaf.