Diplomacy includes the methods and strategies to deal with other Great Mages nonviolently. There are a number of options available here, which all depend on either the exchange of resources, like gold or mana, or military agreements, like non-aggression pacts or alliances.
You can request to trade a specific amount of gold for mana with another Great Mage, or vice-versa. The other mage decides whether or not the trade is favorable enough to accept. The scales icon in the middle shows what the Great Mage thinks of your offer. When your offer is considered fair, or the scales are tipped in their favor, the Great Mage will accept the trade. Sometimes a Great Mage will offer a trade; mages who you are allied with are more likely to do this than other mages.
It is worth keeping in mind that Mana is considered about ten times more valuable than Gold in trades.
You can give gifts to other Great Mages by offering an amount of Gold, Mana or both on your side of the window, and nothing on the opposite side. This serves only to improve relations with this mage, so that they will respond favorably to any offers made in the future. Other Great Mages will never offer gifts, however.
You can demand resources from another Great Mage by selecting an amount of Gold, Mana or both from their side of the window, and nothing from your side. You will only be able to demand as much Gold or Mana as the mage has stockpiled at that moment, which is often very little with most AI opponents. Unlike trades, you have no way of being sure of whether the mage will give in to your demand, or declare war.
Other Great Mages will regularly demand resources from you, which can range to hundreds or even thousands of gold or mana units depending on how much you have stockpiled. The only way to reject these demands is to declare war, which will worsen your relationship with that Great Mage. The good news is, you can offer to make peace immediately in your following turn, and prevent any military action against you.
At any point in your turn you can declare war on any of the opposing Great Mages. This will allow you to take military action against them in the same turn, and also cast Banes such as Counterspell, Locust Swarm or Mana Leech. This is often necessary in long-running games, where other Great Mages will often attempt to cast the Unity spell.
Other Great Mages will only very rarely declare war on you for no reason. In addition, they will always conduct negotiations at the end of their turn, and will never attack you when war has been declared in the same turn.
You can declare war on a Great Mage even if you are presently in a NAP or alliance together, though when you do this you gain the 'Treacherous' modifier with every other Great Mage, reducing your relationship with them by an additional 30 or so. This number will go down over time, but additional offenses bring it back up by another 30 each time.
Declaring war out of an alliance can be devastating to your former ally if played correctly. If you don't care what the other Great Mages think of you, go ahead and park your units around all your allies' towns and then just wait for the right moment. If you position yourself correctly you can swallow up a whole country in one fell swoop.
Also at any point in your turn, you can offer to make peace with a Great Mage you are at war with. Peace offerings can go hand-in-hand with gifts or demands of Gold or Mana to sweeten the deal. A Great Mage who is much less powerful than you will sometimes offer to make peace and give you Gold and Mana as tribute. On the other hand, mages who consider themselves better than you will not accept your peace offering unless you give them a substantial amount of resources as well.
A peace offering can be made at the very end of your turn, which allows you to carry out attacks with all of your units and buildings before giving up your chance to do so. After making peace, you can no longer attack that mage in the same turn.
Non-aggression pact is an option in the diplomacy window with any Great Mage you are currently at peace with. A Great Mage will break a non-agression pact with the player if you lose enough standing with them. A Great Mage will sometimes offer a non-aggression pact to the player, sometimes offering tribute as well. A Great Mage will usually accept a non-aggression pact if you have a respectable standing army or considerable wealth; if you have a negative standing, they'll want resources and generally break it in a turn or two anyway.
If you have a non-agression pact with a Great Mage, you can propose an Alliance with that Great Mage. This usually takes bribery or a high standing with the Great Mage in question, or both. A Great Mage will not usually break an Alliance unless your standing drops considerably. Being in an alliance does not require you to assist your ally in any way, and you may be allied with two, or more, warring Great Mages. You may also be allied with every other Great Mage at the same time. If you cause damage to an ally's unit with an AoE, you will gain the Merciless modifier with that Great Mage, reducing your standing with him by 30.
Break Alliance / Break Non-aggression pact
This option is fairly self-explanatory; breaking either will lower your relationship with the Great Mage in question. Breaking an alliance or NAP will put you at peace with the Great Mage, and breaking an alliance will also give you the 'Treacherous' modifier.