Formal:
* Je suis occupé(e). (This is the most common and straightforward way to say "I am busy.")
Informal:
* Je suis pris(e). (This literally means "I am taken," but it conveys the same meaning as "busy.")
* J'ai beaucoup de choses à faire. (This means "I have a lot to do," and is a good option if you want to emphasize your workload.)
* Je suis débordé(e). (This means "I am overwhelmed," and is a more intense way of saying you are busy.)
More specific:
* Je suis occupé(e) en ce moment. (This means "I am busy right now.")
* Je suis occupé(e) à... (This means "I am busy with...," followed by the task you are occupied with.)
The best option for you will depend on the situation and who you are speaking to.