The first reward you will get is Alliance Points (AP). For partaking in PvP combat or by completing the repeatable quests in Cyrodiil you will receive these. Alliance Points serve different purposes. They are equal to EXP, but are also a currency. Whereas you can buy the basic equipment with in-game gold, you can buy these also with AP, as well as more advanced weapons, gear, motifs and a lot more.
AP are awarded as a separate currency to your character, but can also be stored in the bank and withdrawn on other characters. These do not count as new AP though on another character, so that you can't level up with them on a different character.
As rewards you will obtain a Rewards for the Worthy coffer in your mail for every 20,000 AP earned. The first one of the day will contain a Transmutation Geode that can be split in 4-25 Transmutation Crystals, and will contain a random set item that can only be obtained this way.
You can also receive an Arena Gladiator's Proof from Conquest Missions. Along with the Arena Gladiator items that can be purchased from War Researchers, you can create various Arena Gladiator Runeboxes. These can hold an Emote, a Helm, a Costume, a Pet or various other rewards. You can only earn one Arena Gladiator's Proof per 20 hours per account.
At the end of a campaign you can receive a reward depending on the AP you have earned. You must be under the top 10% players of Cyrodiil to receive a reward up to 19500 gold and 5 Epic items in the 30 Day campaign. Additionally, you will receive a Transmutation Geode that you can shatter into 50 Transmutation Crystals for reaching the minimum of Tier 1 in the 30 Day campaign. You need to reach a minimum of 25,000 AP in order to receive this reward.l