Native Brazilian here.

I will tell you the history as I've learned in school, some 25 years ago. :)

When the court (Royal Family) came from Portugal to Brazil with the help of Britain (due to Napoleon raging war around Europe), they settled in Rio de Janeiro because they felt comfortable around here.

Note that by that time, around 1808, Brazil was a very profitable colony to Portugal, and Rio was one of the major provinces, along with São Paulo (strong in the coffee trade) and Minas Gerais (gold and other valuable resources trade).

Long story short, the court was settled here in Rio, and in an inversion of roles, ruled Portugal from afar. People in Portugal began to fell uncomfortable with that, and demanded that their king returned home with the court. When Dom João decided to go back to Portugal, his son, Dom Pedro, stayed behind and ran the colony as some sort of "sister-state" to Portugal.

As mostly happens in life, strife began about money and ideology, and that led to some some people revolting inside Brazil, claiming they did not need to be a colony any longer. Note that various groups were involved, each pursuing their own benefits (money and power, obviously).

That escalated when Dom João (Dom Pedro's Father, who returned to Portugal some years before) ordered that Dom Pedro also returned home to Portugal and that Brazil ceased all court affairs, remaining as a colony of Portugal.

People did not look at that happily, and then, with the help of those groups mentioned above, on September the 7th of 1822, Dom Pedro proclaimed himself Emperor of Brazil, severing the political bonds (but not that hard) with Portugal.

Some battles occurred, especially in the Southeast and Northeast parts of Brazil.

Note that Brazil had to pay a fee to be "decolonized". The fee was about 2 million sterling pounds, which, at that time, broke the country completely (that and the numerous lending Brazil took from Britain to finance their campaign against Portugal. Oh, and the case that when Dom João sailed back to Portugal, he took all the money he could from Banco do Brasil - "Brazil's Bank" -, the very first bank founded in Brazil by Dom João himself).

EDIT: I am keeping the story very simplified, because it involves a lot of details, names, motivations and parts that would have me writing here an essay :)

If you want to know more (or drill down on details), you can consult the Wikipedia or search on Google.