Apelido vs. alcunha
The difference between alcunha and apelido is primarily geographical. While both relate to names, their usage changes drastically between Brazil and Portugal. In Brazil, apelido is the standard word for meaning nickname, whereas in Portugal, apelido means surname and alcunha is used for nickname.
Apelido
A2In Brazilian Portuguese, apelido is the common, everyday word for a nickname used among friends, family, or colleagues. However, in European Portuguese, apelido refers strictly to a family name (surname), and using it to mean nickname may cause confusion.
O meu apelido de infância era Magricela.
(My childhood nickname was Skinny.)
Eles deram um apelido carinhoso ao novo aluno.
(They gave the new student an affectionate nickname.)
Ela não gosta quando a chamam pelo apelido.
(She does not like when they call her by her nickname.)
O jogador de futebol é conhecido pelo seu apelido.
(The soccer player is known by his nickname.)
Alcunha
B2In European Portuguese, alcunha is the standard word for a nickname. In Brazilian Portuguese, alcunha is a formal, old-fashioned, or specific term; it is often used in legal contexts to describe a criminal alias, a historical epithet, or sometimes a derogatory label.
Naquela aldeia portuguesa, todos se tratam pela alcunha.
(In that Portuguese village, everyone addresses each other by their nickname.)
O rei D. Pedro IV tinha a alcunha de O Rei Soldado.
(King Pedro IV had the moniker The Soldier King.)
O suspeito era procurado pela polícia sob a alcunha de Louro.
(The suspect was wanted by the police under the alias Blondie.)
Colocar uma alcunha ofensiva em alguém é falta de educação.
(Giving someone an offensive nickname is bad manners.)
Summary
To summarize, if you are in Brazil, use apelido for nickname. If you are in Portugal, use alcunha for nickname, because there apelido means surname. In Brazil, alcunha implies a formal alias or moniker, while in Portugal it is the casual term.