Whenever you have any questions like this it is very important to always look at the word in context to see if there are any contextual clues that might give a hint as to the meaning of the word you are trying to identify. This is a very important skill to develop as often, even when we have no idea what the word we are looking at means, we can deduce the meaning from its context and the other words surrounding it. Let us apply this process in this case:
And then, in a twinkling, I heard on the roof
The prancing and pawing of each little hoof.
As I drew in my head, and was turning around,
Down the chimney St. Nicholas came with a bound.
The key phrases to focus on therefore are "Down the chimney." How does St. Nicholas go down the chimney? The answer is "with a bound." We can therefore deduce that "bound" means "leap." This of course fits our general impression of St. Nicholas.