The Ender Chest is a special Chest that stores its inventory in The End, keeping your items everywhere and nowhere at the same time. Each chest can access the same inventory allowing you to transport items across vast distances instantaneously. It even works between dimensions. Each chest has a code represented by the three wool pads on the top of the chest. Any chests with the same code will be linked.
If there is more than one Ender chest on a map, when ever it is opened the other Ender chest lid will pop up as well.
How to Use
Each of the three colour pads located on the chest's top face can be recoloured by right-clicking a dye on the pad or by using the following recipe with the three dyes in the crafting table.
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 (16*16*16) or 4,096 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 mined using a pickaxe (preferably diamond). You cannot change the code or break a chest that is currently open.
Ender Chests v.s. Teleport Pipes
An Ender Chest is worth approximately 7424 EMC in materials, this makes it a better choice than using Teleport Pipes, as it is slightly cheaper than the 8192 EMC per teleport pipe. Ender Chests are also more reliable than teleport pipes because while Ender Chests have no bugs, teleport pipes have more bugs that will crash your game/server if they occur.
Ender Chests v.s. Vanilla Ender Chests
It is unknown what happens when a wooden pipe is connected to a Vanilla Ender Chest (Vender Chest for short), because it is unknown which player inventory the wooden pipe will come from.
It is also unknown what happens when a pipe, any pipe that places items into a chest or a machine, is connected to a Vender Chest, because it is unknown which player inventory the pipe will pipe the items to.
The fact of the matter is is that we won't know these things for sure until Buildcraft (as Buildcraft, not tekkit, brings pipes) updates to the Minecraft version with Vender Chests, and even then tekkit will likely wait for all the mods to update. So the following comparison is speculation and unproven, but it represents the most likely behavior of the Vender Chest in Tekkit.
As stated above Ender Chests are worth 7424 EMC, which makes using V Ender Chests far cheaper with an EMC value of 2034 (8 obsidian * 64EMC + 1 eye of Ender * 1,792EMC = 2034). However, V Ender Chests are player specific, meaning that all players on a server will see different inventories when they open a V Ender Chest, whereas Ender Chests have only server-specific inventories, meaning that all players on a server will see the same inventory when they open this chest.
This player-specific inventory makes using V Ender Chests instead of Ender Chests on a server where there is no pooling of resources a very good idea, because other players will be unable to access your resources in the V Ender Chest due to the V Ender Chest's inventory being linked with only you.
On the other hand, this player-specific inventory makes using V Ender Chests instead of Ender Chests a very bad idea if you are playing on a server where everyone works together and pools their resources, or work on a team at the very least, because the player that the V Ender Chest's inventory is linked to must manually take out the resources from the V Ender Chest in order for the rest of the players to have access to them.
In Game Usage
You can take a chest with you to the nether, place it, put your nether ores and glowstone safely within as soon as you mine them, and empty your inventory without having to return all the way to the portal. You may also use enderpouches (remember to change their colour) (Note: an Alchemy Bag also works.)
It can be used to share items between multiple bases. You can store all the mine carts for your railway stations in the one chest and have them accessible from any station. You could have a public food chest, and everyone on the server could access it at the same time.
The Ender Chest coupled with the Ender Pouch and Buildcraft Pipes can be used to make a player's inventory infinite. When you place items in an Ender Pouch, the items are also placed in the linked Ender Chest. By connecting a pipe system to the linked ender chest that will pump items from said ender chest to a different chest, then the ender pouch will never fill up, because every item the wooden pipe of the pipe system pumps out of the linked ender chest disapears from the ender pouch. This infinite inventory solution is useful if you are mass mining blocks--ore, stone, dirt, sand, etc.--and don't care whether or not you keep them in your invenory. To see how this method of obtaining infinite inventory compares to Equivelent Exchange's method use the Ender Pouch link provided in the first sentence of this paragraph and scroll down to the category that says "Ender Pouch v.s. Alchemy Bag".
As said in "Ender Chests v.s. Teleport Pipes" category, Ender Chests can be used as a cheaper and more reliable alternative to transport/teleport pipes. Also doesn't cause lag like a long chain of transport pipes does.
If one chest is opened all chests with the same code will be opened.
In Minecraft 1.3.1, a similar Item, also named Ender Chest, was released to Vanilla (Default) Minecraft but with some different features. It does not have the Colored Wool at the top and is player specific. So if you open one it will be empty (until you put something in it) no matter how much other people put in it. (not yet released on Tekkit)