Learn Hardcore Polish: Mój wesoły brat uważa, że piłka ręczna to najlepszy sport. - My cheerful brother thinks that handball is the best sport.

Because brat (‘brother’) is masculine in Polish. That means both the possessive pronoun mój (‘my’) and the adjective wesoły (‘cheerful’) must agree with a masculine-singular noun in the nominative case. If you were talking about a sister (siostra, feminine), you’d say moja wesoła siostra instead.

What does the verb phrase uważa, że mean and how is it used?

Uważa, że comes from the verb uważać, which here means ‘to consider’ or ‘to think’. When you want to express an opinion, you use uważać, że + a subordinate clause. For example:
Uważam, że to dobry pomysł. (‘I think that’s a good idea.’)
In our sentence, Mój wesoły brat uważa, że… means ‘My cheerful brother thinks that…’.

Why is there a comma before że in this sentence?

In Polish, subordinate clauses introduced by że (‘that’) must be separated from the main clause by a comma. So you always write
Mój wesoły brat uważa, że piłka ręczna…
just like English often has a comma before “that” (although English sometimes omits it).

What is the structure of piłka ręczna and why is it not ręczna piłka?

Piłka means ‘ball’ and ręczna is the adjective ‘hand-’ (feminine form). In Polish sport names of this type the head noun (piłka) typically comes first, followed by its qualifier (ręczna): piłka ręczna (‘handball’). While ręczna piłka would literally mean ‘a hand ball’, it’s not the standard fixed expression for the sport.

Why do we use to instead of jest in piłka ręczna to najlepszy sport?

Here to acts as a copula in a nominative-to-nominative construction:
X to Y
Both X and Y remain in the nominative case. It’s a common, slightly more conversational alternative to using jest + instrumental. If you prefer the verb jest, you must change the predicate to the instrumental:
Piłka ręczna jest najlepszym sportem.

Why is najlepszy sport in the nominative case after to?

Because in the X to Y pattern neither side takes an instrumental—they both stay nominative. To is not a full verb but a particle linking two noun phrases. If you switch to a full verb (jest), then the second noun phrase must go into the instrumental case.

How is the superlative najlepszy formed?

Najlepszy is the superlative of dobry (‘good’). It’s irregular: the comparative of dobry is lepszy (‘better’), and you add naj- to form the superlative najlepszy (‘best’). Other adjectives usually form the superlative by attaching naj- to the regular comparative:
wysoki → wyższy → najwyższy.