I found it confusing having two verbs for to wait.
Waiting, in Portuguese - parsing the possibilities
What is the difference?
When talking about waiting you won't find a hard rule for either of these verbs. Look at these examples:
Do you see the difference?
Brazilians don't even think about the difference but language hijackers like us do.
Esperar is the everyday, conversational way to say to wait.
Aguardar is more formal. (emails, more formal situations)
Let's do another:
Formal situation. I'm imagining someone waiting or, GUARDING for this decision.
Here's another way that AGUARDAR is used:
This is classic of business communication. Think of "fico no aguardo" as, to stay in wait for something.
And here's another way that ESPERAR (specifically) is used:
It's important to remember that only esperar also has the meaning of, to hope for.
Hope that helps.
Espero que seja útil!