How many of each structure can I build in clash of clans.
It depends on the building in question and your town hall level.
For instance, for Cannons, you can build 2 until town hall 5, when you can build a third. Town hall 7 adds 2 more for a total of 5, and you get one more at town hall 10 for a total of 6.
There is a table on this Wikia page which lists the buildings available by town hall level.