Diplomats are emissaries from different civilizations that visit your fortress. They typically arrive shortly before or after the caravan would normally show up.
Human diplomats come when your fortress becomes important enough to rate a baron or better noble. This individual will only speak to your baron (or better), and generally says something inane such as "It's such a pleasant place you've carved out for yourselves." As the noble the diplomat wants to speak with never does anything anyway, you need do nothing to ensure the meeting occurs. If you're in a state of war with the human civilization, the diplomat may come proposing a peace treaty.
The Elves will eventually send a diplomat who meets with your head noble (such as the Duke) to talk about the "tree situation" at your fort. The Elven diplomat's first arrival will be completely unannounced (i.e. the game will not pause and zoom to the map edge) and will consist either of a stern warning (if you have cut down any trees) or a congratulation for not disrespecting the forest. Subsequent visits will be announced as usual and he will demand that you limit the number of trees you cut down before the elves return the next year. This count will be visible in your ivilization list (view the Elven civilization and press several times) and will decrement for each "Fell Tree" job your wood cutters complete, including subterranean tower-caps. If you stay within the quota, diplomatic bonds with the Elves will be strengthened, while violating the quota can strain relations and potentially lead to war.