Discar vs. marcar
Both discar and marcar can mean to dial a telephone number. The difference lies primarily in their origin and regional preference. Discar comes from the word for disk (referring to old rotary phones) and is the technical standard, especially in Brazil. Marcar technically means to mark or to press, referring to pressing buttons, and is the preferred term for dialing in European Portuguese.
Discar
B1This verb specifically refers to the act of inputting a telephone number. It originates from the era of rotary phones where one had to dial a disk. It is widely used in Brazil and is the standard word used in automated customer service systems.
Por favor, disque zero para falar com a operadora.
(Please, dial zero to speak with the operator.)
Eu acho que disquei o número errado.
(I think I dialed the wrong number.)
Você precisa discar o código de área antes do número.
(You need to dial the area code before the number.)
Ele começou a discar, mas depois desistiu da chamada.
(He started to dial, but then gave up on the call.)
Marcar
A2While this word often means to schedule or to score, in the context of phones, it means to dial by pressing keys or buttons. It is the dominant term used for dialing in Portugal (European Portuguese), whereas discar might sound slightly more technical or old-fashioned there.
Marque o número no teclado do telemóvel.
(Dial the number on the mobile phone keypad.)
Se você marcar esse número, ninguém vai atender.
(If you dial that number, nobody will answer.)
Tentei marcar o número dela, mas estava ocupado.
(I tried to dial her number, but it was busy.)
É mais fácil marcar os números na tela grande.
(It is easier to dial the numbers on the big screen.)
Summary
To summarize, use discar when you want to sound technically precise or are in a Brazilian context; it recalls the action of rotating a dial but applies to keypads too. Use marcar specifically for the action of pressing the keys to compose the number, especially if you are speaking European Portuguese.