What do you do if you forget to acknowledge someone's birthday?

I do one or more of what you named above. It depends whose birthday I forgot and how much time passed when I finally remembered.

I send a card or a note if it is someone I have exchanged birthday greetings with in the past. If not part of a regular exchange over the years, I will let it slide.

one word: belated

Send a belated birthday card and maybe a text, email for Facebook message saying to look for the card, and a cute sorry for forgetting. Nothing dramatic.

If I forget the birthday of someone I’m very close to then I might send my apologies in addition to my regards. Usually, though, I just send a belated message over e-mail or messenger/text. I very rarely send cards through snail mail. I love snail mail! It’s just not as expedient.

I don’t have many people expecting me to acknowledge their birthdays. When I do, I give them a call and talk to them and mention that I forgot to call them on their birthday…

It depends, I might give them a call instead.

Better late than not at all.

Tell them later.

Send a belated card.
And move on.