In order to open a chest/coffer you need either the chest/coffer key for that zone, or you need to be a
thief with lockpicking tools (see: Picking your Coffers and Chests).
After being opened, the chest or coffer will respawn in one of the designated locations after approximately five minutes; there are two exceptions to this, both of which cause the chest/coffer to respawn immediately:
If a chest is opened and a Magical Pattern is obtained (for use in the quest The Goblin Tailor)
A Thief attempts to pick the lock of a coffer and fails, producing a Mimic.
Treasure Chest and Treasure Coffer for more details.
Treasure Chests can be found in the lower level dungeons. They can contain some good items and give decent gil.
Treasure Coffers are generally found in higher level dungeons and contain better items and generally more gil.