Diplomacy isn't all that powerful against an AI opponent. The real power is in making treaties, war, trading, or demanding tribute.
Treaties
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) or Declaration of War. 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.
Being stronger in terms of power in Civilization III usually helps your diplomatic stance in some ways, even if it makes some of the other civilizations suspicious of your motives. For one thing, you no longer have to deal with irritating Romans/Zulus hailing you and demanding stuff.
Speaking of those demands, be aware that the comps will do stuff like that to try and bait you into getting into a war when they are paying you a lot of money as part of a trade.