An apiary is a house for bees, which is used to make Honey.
You will need to purchase an Apiary from the general store or the catalog and place a Queen Bee in it to start the honey making process.
The Queen will need to rally 10 Bees before Honey production begins. She will gather one each day or you can speed this up by placing worker bees into the apiary.
It takes 5 days to produce Honey once the Queen and her 10 bees start. You will then need a smoker to spray the apiary to calm the bees down and an empty Jar to collect the honey.
The placement of the apiary does not appear to matter, meaning you currently don't need to place it near flowers.