Getting the Core
An 
Ender Core is the item that powers a 
Teleporter. You obtain it by crafting, using ender-related materials as the central ingredient surrounded by the mod's required components. Check the recipe in JEI or REI in-game, since the exact layout is shown there and is the reliable source. Once crafted, the Ender Core sits in your inventory until you install it into a Teleporter block.
Installing It
Hold the Ender Core and right-click a placed Teleporter. The core is consumed from your hand and stored inside the block. A Teleporter without a core cannot be given a destination, so this step always comes first. If you break a Teleporter that already contains a core, the core is returned to you, so you can move your network around without losing materials.
When to Use the Enhanced Version
The 
Enhanced Ender Core is the upgraded variant. It is used where a plain Ender Core is not enough — for longer-range links or for Teleporters that need to reach across dimensions, depending on how your configuration is set. Treat the Enhanced core as the option for ambitious networks and the basic Ender Core as the standard choice for same-area travel. Always confirm the current recipe and behavior in-game rather than assuming, since configuration can change what each core supports.