Learn Portuguese (Portugal): Tenho uma consulta às três.

Às is a contraction of the preposition a (at) and the feminine plural article as (the). We use the feminine plural because it secretly refers to the feminine plural word horas (hours). You are literally saying 'at the (hours of) three'.

Why isn't there a word for 'o'clock' or 'hours' after 'três'?

In everyday spoken Portuguese, you do not need to add the word horas when stating a time exactly on the hour. Just saying às três is the natural way to say 'at three o'clock'.

Does 'consulta' mean any kind of appointment?

No, a consulta refers specifically to a medical or professional appointment, like seeing a doctor or a dentist. If you were meeting a friend, you would use encontro, and for a business appointment, you would use reunião.

How do we know if 'às três' means 3 AM or 3 PM?

Context usually makes it clear (a doctor's appointment won't be at 3 AM!). In casual speech, Portuguese speakers use the 12-hour clock. In formal schedules or written text, they often use the 24-hour clock, so 3 PM would be written or spoken as às quinze (at fifteen).