Diplomacy isn't all that powerful against an AI opponent. The real power is in making treaties, war, trading, or demanding tribute.
When you start the game you are all alone with no one around you, but as you grow and expand your nation you find neighboring civilizations. These civs can either be your ally or foe. Obviously you can destroy him or her, or you can make peace, which is probably a good choice. Early in the game you can only make peace treaties with warring nations. This itself is not very effective because you can break that treaty at any time that you wish. As you progress in the game you can build Embassies. These embassies allow you to make "advanced" treaties such as Right of Passage (RoP). The RoP allows your and your allies units to freely traverse each others' terrain. You don't exactly have to be an ally to make a RoP, actually you can have one with a nation that you don't like
Some additional diplomatic tools that appear later in the game include economic embargoes and Mutual Defense Pacts. The former is simply an agreement to not trade anything with a third-party civilization, and the latter is an alliance that obligates you to come to the defense of the civ you are allied with when they get themselves into war. Be aware that the latter is really a two-edged sword; while having the alliance can be nice, you can easily get dragged into another civilization's wars. This is a particular problem on the higher difficulty levels, where the computers are almost always at war and frequently with civs they are no match for. Be careful not to let yourself get drawn into a fight with a faction that will kill you.
Being the strongest or stronger than most of your rivals is usually a plus in Civ III, since civilizations will not be inclined to either start wars with you at random, or call up demanding things (a particular issue with some of the war-like computers, such as the Romans or Zulus). However, the downside is that nearly all of the rival civs will be suspicious of your motives.
One crucial hint regarding trade - you may find yourself in a position where the computers are offering significant per-turn tribute, particularly if you are a highly advanced civilization selling technology. This can swell your treasury, but it has a significant downside; when the computers begin to run short of cash, they'll call up and make demands for you to refuse, in the hopes of getting an opportunity to declare war and end the tribute payments. Be aware of this, and be very careful not to rise to their bait if the agreement is particularly lucrative.
That said, the above isn't a perfect solution, since the enemy civ may decide to simply declare war on you regardless. Force them to do this, then rally your friends and fight them off.
Another key hint about Diplomacy: be very careful when ending a war if you have one or more Military Alliances against a faction. Calling up the enemy and making peace ahead of your allies is seen in a very bad light, and usually will send a Civ's attitude down to "Furious", even if the entire campaign they were in a "Gracious" mood. If possible, try to get the other civs in your allegiance to declare peace first, then finish up yourself if you are not desperate for peace.