To buy a building tap the Shop icon in the lower right corner of the screen.

Buildings are divided into categories:

Choose a category and tap it once to open up a list of available structures. You can swipe the list right and left to access all structures in a category. Some of them may be locked until certain requirements are met, like achieving a higher level or increasing your town's population.
To place a certain building in your town tap the green button below it.

If you have enough coins to construct it, the Shop window will close, and your building will appear on your screen in the Editing mode.

Editing mode allows you to Move and Rotate the building before you actually place it in your town. While in this mode, tap the round arrow button to rotate the building or tap and slide the structure itself to relocate it. When you're done, tap the check mark button to apply changes.
Some buildings go up in stages, so it may require a few more taps to construct.