The imperials as stated by their name own the empire. I don't know how much of the the territory outside Cyrodiil they technically own, but some if not all of the territory of the three factions belongs to the empire. Unfortunately with a big Daedric anchor in the imperial city they have lost control of the empire.
Some, if not most have sided with the worm cult.
The three alliances are fighting the worm cult, and in turn the imperials, that's your PvE. Each alliance is also taking advantage of the current state of empire to take Cyrodiil for themselves, and take control of the empire. With a civil war raging in Cyrodiil for its ownership that is your PvP.
(This is what I've picked up along the way as I've played, anyone who knows their lore better please feel free to correct me).

The empire does regain stability. If you can see a map from the game Oblivion set 800 years later in Cyrodiil you will see their territories, many of them playable in ESO. At this point the empirebis doing well, until the emperor is assassinated at the beginning of the game, marking the decline of the empire.

In the game Skyrim 1000 years after ESO the empire has become a toothless dog. Civil war looms in Skyrim between the empire and the local Jarls that support the empire, against a Jarl who has formed a militia against the empire, and declared himself high king of Skyrim. Due a complex set of circumstances, the the dominion have enforced a ban of the worship of Talos throughout the empire, and for Nords he is their living god. The dominion claim a human cannot be a god. The empire is weak and is being made to enforce this rule. The rebels wand independence, and the right to follow the 9, (on ESO you will hear of the 8, the nine are the same but with talos included). The empire I stuck between a rock and a hard place, throughout the game you feel this is probably the final days of the empire.