In typical Hemingway fashion, a lot of what we know about these characters comes from their dialogue. We don't really get anything in the way of description – in fact, pretty much all we know about both the old man and the younger waiter comes via conversation. Though the dialogue, which goes largely unlabeled at first (we're not sure which of the two waiters is speaking), is initially kind of hard to follow, we quickly learn to differentiate between the younger waiter and the older waiter. The younger waiter's selfish desires come out clearly in his statements; he doesn't care about the loneliness or unhappiness of the old man, and simply wants to get home to his waiting wife. The older waiter, on the other hand, is sympathetic to the old man, and his musings reflect a certain understanding of what the old man wants from the café – he sees it in himself.
We only really see into the thoughts of one character, the older waiter. However, in this brief glimpse into his mind, we glean a great deal of understanding; without letting us "in" for just a moment, Hemingway's story would have been a lot less interesting, and a lot less accessible. In our brief foray into the older waiter's thoughts, we see a continuation of his conversation with the younger waiter, in which he mentally comments on what he sees as the emptiness of life – "nada y pues nada" (nothing and then nothing), a statement that helps the rest of the events of the story fall into place for us.
This is plain and simple: there aren't any. The characters here are simply identified by their traits or jobs – the old man, the older waiter, the younger waiter, the soldier – and we don't get much more personal information about them. While they're certainly individual characters, this choice implies a kind of universality about them; for example, we feel like this isn't one specific old man in Spain, it could be any old man out there in the world.