The core is a 3x3 block you cannot currently build and a part of the starting blocks. The core is your main base and resource storage. The enemy (bots) goal is to destroy the core and cause as much damage as possible on the way. You can input into all 12 tiles on its sides with any transportation block however there is no way to get items out of the core, so non-construction items (fuel; coal and uranium) will be unusable in the core. The player spawn is the bottom middle tile next to the core, you are unable to build blocks on this tile that pc players cannot walk over. The core has 800 health which you can view by selecting it on mobile, and hovering over on PC. It is useful to know that Repair turrets will repair the core.
It is also possible to view the exact core health with the use of the debug console. There it is labeled as CORE.HEALTH.
When your core is destroyed, your screen will automatically scroll to the core's location and you watch the final explosion animation. Your screen then darkens with a bright dialog in the center saying "YOUR CORE HAS BEEN DESTROYED" and also what wave you lasted to.