NOTE: This is not to be mistaken for the SMP ender chest, added in vanilla minecraft. Though they are similar, they cannot operate together in any way; they don't transport together.
The Ender Chest is a special Chest added by the Ender Storage mod. Each chest has a code represented by the three wool pads on the top of the chest. Chests with the same colour code will share the same inventory.
If there is more than one Ender chest on a map, when ever it is opened the other Ender chest lids will pop up as well.
Each of the three colour pads located on the chest's top face can be recoloured by right-clicking a dye on the pad. Placing a Diamond on the latch of the Ender Chest will convert the chest into use for your own network of Ender Chests, useful in SMP.
Note that since there are 16 dyes to choose from, and you can use the same colour dye on more than one pad, in total there are 4,096 (16*16*16) possible colour combinations. This makes it very unlikely (but not impossible) that you will pick the same combination as (or have your code guessed by) another player. It would take a significant amount of time and dyes for a player to systematically check all possible combinations in order to raid your chest.
As they are made of obsidian, Ender Chests are picked up using a pickaxe (preferably diamond). You cannot change the code or break a chest that is currently open.
Using the Ender chest and an ender bag with the same colour combination linked to a sorting machine, you can make a near never ending size bag which will automatically be transported back to your base, via the end!