In the pics above, the green area with the path down the middle is the outer ward, which leads through a gateway - originally there was a drawbridge - into the inner ward, where the royal apartments were. The fourth pic down is the well, 91 feet deep.
Construction began on Conwy in 1283; for the history of the castle, take a look at the page here. Interesting Conwy fact: in January 1326, Edward II appointed Aline, Lady Burnell, constable of the castle. It was most unusual for a woman to be put in charge of such an important stronghold, though no doubt the fact that Aline was Hugh Despenser the Younger's sister was a major factor in Edward's choice.
The name comes from the Anglo- Norman beau mareys, 'fair marsh'. For the castle's history - it was begun in 1295, and never finished - see here.