A chest is one of the most common place to loot for items. Most NPCs won't mind you looting a chest next to them or inside their homes or taverns. Objects like chests shimmer (at default settings) when they contain something you can loot. Some chests are locked and you will need either a key or sufficient lockpick skill to open it.
Trivia
- If you try to open the chest near the keeper of the Dalish elves, his apprentice will yell at you, if you do it again she will become angry.