There are currently six different types of Chests and their respective Keys:
They are all found in the Chests menu, and each has its own section as shown below.
Chests that are purchasable with Gems (Gem, Event and VIP Chests) are offered at a discount of 5% or 10% if opened 10 or 50 at a time, respectively.
Gold Chests are the lowest-quality chests and offer both the least variety and the least value in their contents - only Common and Rare troops and traitstones can be found in these chests, as well as some Gold, Souls, Glory and Gems. They are also the easiest chests to unlock, since Gold Keys can be purchased with Gold (300 each). It is worth noting that this is currently one of the only ways to get Common troops.
Glory Chests are the bread-and-butter of chests. They are almost as common as Gold Chests but offer the widest range of content - any troops Rare or higher can be found, all the way up to the occasional Legendary or Mythic troop, as well as the same basic resources carried over from Gold Chests. As the name suggests, Glory Keys can be purchased with Glory (20 each).
Gem Chests are the highest-quality of the more basic chests. A step further than Glory Chests, Gem Chests don't offer Rare troops or traitstones - the lowest rarity in a Gem Chest is an Ultra-Rare - and also don't offer basic resources. There is a correspondingly higher chance of getting even rarer troops - reportedly 4x for Legendary and 10x for Mythic. Gem Keys can be purchased with Gems (10 each).
Event Chests are similar to Gem Chests in their rewards, but have the unique property of being tied to the current weekly event in terms of what specific troops can be found. For instance, if the event is in Broken Spire, the troops in Event Chests will largely be from Broken Spire (completely so for Epic or higher). This comes at a cost - Event Keys are more expensive than Gem Chests at 15 Gems each.
Guild Chests are the most complicated of chests (see this article). They are unique in that their contents' value increases as a guild's accumulated Guild Seals increase during a week - at peak value, their contents mirror those of Gem Chests. Additionally, the guild chests have around a 50% chance to yield Guild Guardian troops, which can't be obtained through any other means. Guild Keys can be earned from Guild purchases or by spending 20 Guild Seals per Key.
VIP Chests are unique in that they are only available to players who have earned a VIP Level of 5 or higher. They only have contents of Epic quality or higher, with the added bonus that Epic quality items and Celestial Traitstones are doubled (literally doubled, two items instead of one). Each VIP Chest also comes with 2000 Gold. VIP Keys cost 50 Gems each.
Chest Contents Chart
(Note that the values for Guild Chests are assuming Level 6 or maximum value - lower levels can have lesser quality content. Again, see this article.)
*: 2000 Gold per Chest
**: Double Epic Troops, Arcane and Celestial Traitstones