General:
* Short pants
* Breeches
* Knickerbockers (more old-fashioned)
* Bloomers (typically associated with women's clothing)
* Trousers (though this usually refers to long pants, it can sometimes be used for shorts, especially in British English)
More specific:
* Cut-offs (for denim shorts)
* Boardshorts (for surfing or swimming)
* Cycling shorts
* Running shorts
* Cargo shorts
* Hotpants (for very short shorts)
Informal:
* Sorts (slang)
* Playshorts
* Walking shorts
The best synonym will depend on the specific situation and the desired tone.