General:
* Top
* Blouse (for women)
* Tunic (longer shirt)
* Polo (collared shirt with a button placket)
* Sweater (knitted shirt)
Formal:
* Dress shirt
* Tuxedo shirt
* Vest (formal shirt without sleeves)
Informal:
* Tee (short for T-shirt)
* Tank top (sleeveless shirt)
* Undershirt (worn under other shirts)
* Vest (informal shirt without sleeves)
Specific styles:
* Hawaiian shirt
* Flannel
* Hoodie
* Sweatshirt
* Jersey (athletic shirt)
The best word to use will depend on the specific context.