TL;DR
- Winter chests = Normal chests with snow and candy. Same rules apply
- Snowdown/Legacy chests are obtainable through RP Purchase only
- Poro King's treasure chest only obtainable through parts dropped by purchasable chests.
The only chest you can obtain by playing the game is the "Winter Chest". This is basically the same as a normal chest. The only differences are the snow on top and the chance to drop a Snowdown Sweet which can be used to purchase snowdown themed summoner icons etc.
For this chest the standard rules apply. This means Winter chests count towards the normal chest cap and you need an S- or higher on a champion to earn one. You'll also need a key to open it. You can also earn them in any mode where you can earn normal chests.
The other chests are buyable with RP and cannot be obtained by any other means. They do not require a key to open and have better rewards and a chance to drop a Poro King treasure chest fragment.
Poro King treasure chest fragments can be used to craft a Poro King chest which rewards you with a permanent skin shard. These chests also don't need a key to open.