It's an onomatopoeic term for briefly sounding a horn, but I almost never hear it used for that. In my experience, there are two primary uses of the word in American English:

1) In the expression toot [one's] own horn, which means to praise oneself. Often in the form "I don't want to toot my own horn, but [...]".

2) A cutesy word for a fart, used primarily when talking to young children.

Notably, an elephant using its trunk to make noise is said to be trumpeting, not tooting.