Here's an explanation in plain English.
The world is being ravaged by Aragami. Aragami are horrible creatures of drastically varying size, shape, and attributes made out of a material that drives them to consume anything in sight. Some resemble animals or people, some have even grown weapons like tank mortars and armor. They're impervious to conventional weaponry, as they can only be damaged by other Aragami, and even then they seem to be capable or resurrection due to their Core, which remains alive even when the Aragami has been beaten.
The world has been mostly eaten by the Aragami, but a small government remains.
The God Eaters are warriors who are given special weapons that allow them to combat the Aragami. They're generally stronger and faster than normal human beings. God Eaters wear big metal bracelets on their dominant hands that have been attached to their wrist, binding them permanently to their weapon, the God Arc.
God Arcs, of if you prefer, Jinki, are the weapons that God Eaters use. They can only work with the person they've bonded to, and will literally consume anyone else that tries to use them. God Arcs are typically large swords, spears, or hammers, that can transform into guns. They also have built in shields that can protect you from harm. There are older model God Arcs that remain swords or guns at all times. God Arcs are usually pretty big and heavy, but God Eaters seem to be able to swing them around and carry them with no problem.
God Arcs, or if you prefer, at their core, are made of the same crap that Aragami are made of, so the weapons, in order to function properly, must be fed, a process which is typically lethal to enemy Aragami. The God Eater uses their God Arc to beat an Aragami into submission, and before it can regenerate, the God Eater allows the God Arc to grow a big, gnarly mouth and bite the crap out of the Aragami, tearing out its core and killing it.
So, the God Eaters are tasked with surveying the land, hunting down Aragami, and defending areas that are under attack. They are humanity's last line of defense, as most people are living in cities fairly safe, or on the run.