Learn Polish: Zawsze daję bratu trochę pieniędzy. - I always give the brother a little money.

Bratu is in the Dative case. The verb dawać (to give) needs the Dative case to show who is receiving the action. While most masculine nouns take the -owi ending in the Dative singular, brat is a common exception that takes the short ending -u.

Why is the word for money in the form pieniędzy instead of the basic form pieniądze?

Words that express quantity, like trochę (a little) or dużo (a lot), force the noun that follows them into the Genitive case. Pieniędzy is the Genitive form of pieniądze.

Is pieniędzy singular or plural here?

It is plural! Unlike in English where money is treated as a singular concept, the Polish word pieniądze is only ever used in the plural. Therefore, pieniędzy is specifically the Genitive plural form.

Could I use the perfective verb form dam instead of daję?

Not in this sentence. The word zawsze (always) tells us this is a repeated, habitual action. In Polish, you must use imperfective verbs like dawać (which conjugates to daję here) for habits and routines.