Waiting, in Portuguese

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:

Eu vou esperar você voltar ao hotel. > I'm going to wait for you to return to the hotel.

Eu vou aguardar a sua volta ao Brasil. > I'm going to wait for your return to Brazil.

Or this:

Espero você em casa. > I'll wait for you at home.

Aguardo sua chegada. > I'll wait for your return.

Do you see the difference?
It's subtle.
Brazilians don't even think about the difference but language hijackers like us do.

The "RULE":
Esperar is the everyday, conversational way to say to wait.
Aguardar is more formal. (emails, more formal situations)

Let's do another:

Vamos aguardar pela decisão do tribunal. > Let's wait for the court's decision.

Formal situation. I'm imagining someone waiting or, GUARDING for this decision.

Vamos esperar até o final de semana para decidir. > Let's wait until the weekend to decide.

Not formal.

Here's another way that AGUARDAR is used:

Desde já agradeço a atenção e fico no aguardo de um retorno. ➜ Thanks in advance for your attention and I look forward to a return.

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:

Espero que tudo dê certo. > I hope that everything works out.

It's important to remember that only esperar also has the meaning of, to hope for.

Hope that helps.
Espero que seja útil!