This answer is updated as of the Reaper of Souls expansion.
While it's important to realize the random nature of the game and it's drops, here are some general observations I have made in my playthroughs. Keep in mind, your experience may vary:
Resplendent and Exquisite chests have dropped by far the most
gold/gems/rare loot in my playthroughs. It is common for me to find a couple rares, a good loot haul, a couple gems, and very occasionally a legendary item. Of all the chests, these are the only ones that I have seen a consistent increase in the quality and quantity of drops to conclusively distinguish them from any other chest. Couple that with the special sound effects they make when opened, and the fact they appear much more infrequently, and one can discern they have elevated loot.
Cursed Chests operate in the exact same manner as resplendent and exquisite chests once the curse on them has been lifted. When you find a cursed chest, there is generally a short event you have to fight your way through to lift the curse. The events range from killing an elite mob that spawns nearby, killing a certain number of enemies within a time limit, or simply surviving a a few waves of enemies. In some cases, you can improve your loot by meeting certain pre-requisites, which will be outlined for you in your quest pane.
Normal Chests are all other containers that behave as a chest would. For example, bloody chests, treasure chests, demonic vessels, etc. are considered "chests" for game purposes and have random loot. Depending on your level, you will generally get one or two pieces of gear of varying quality, as well as a modest gold haul. Sometimes you will get gems. It should be noted however, that like any container, even a normal chest has a chance to drop legendary items.
Important to remember though, if you can interact with it, CLICK IT. You never know what will come out. On multiple occasions I have received a rare or even a legendary item from an innocuous environmental object or a large sum of gold from a dead soldier. If you value loot, it pays to be a completionist. Explore every nook and cranny of each map and kill or demo everything. It will keep you ahead of the leveling curve, and ensure you maximize earning potential in each area.