Formal:
* Ocupado: This is the most common and general translation for "busy."
* Engajado: Means "engaged" and implies being occupied with something specific.
Informal:
* Atarefado: This translates as "hurried" or "rushed" and implies being busy with a lot of things.
* Cheio de trabalho: Literally means "full of work," indicating someone who is very busy.
Here are some examples:
* Estou ocupado agora. (I am busy now.)
* Ele está muito engajado com o projeto. (He is very engaged with the project.)
* Estou atarefado hoje. (I am busy today.)
* Estou cheio de trabalho. (I am swamped with work.)
The best translation will depend on the specific context and level of formality.