There are five levels of “request”, as I recall (the wiki no doubt has exact details, and there are many forum posts from Steve to read). Each one represents a higher level of importance that the race holds for the system. Each one is accompanied by a deduction of points from that race’s opinion of your race. The amount of the deduction is determined by the level of importance, but also by the tonnage of the ship(s) that they are sending the messages to, and whether or not ships of those ship’s class have ever been observed by the other race to fire any weapons. Edit: specifically, they will completely ignore a single ship provided it has a diplomacy module, it has civilian engines, they don’t know it has weapons, it is less than 10,000 tons, and it’s not in a particularly valuable system. If it is in a valuable system, but all the other conditions are met (or the race has only one system) then they won’t ignore it but they will only penalize you 10% of normal. Xenophobia does also matter, as a xenophobic race automatically rates all ships as being more threatening than a non–xenophobic race.
Simultaneously, your ships with diplomacy modules are sending reassuring platitudes back, which raise their opinion of your race. The amount that their opinion is increased is mostly related to the diplo ship’s commander’s diplomacy skill, plus a random factor. There are no messages at all that this is happening. I have in the past suggested that the ambassador should occasionally send a progress report back to indicate to the player that he is doing something, but to no avail. Edit: their xenophobia also reduces the amount that their opinion is increased.
If you manage to raise their opinion of you to +200, they will ask to sign a non–aggression pact (higher level treaties do exist, so don’t stop there). If it goes below −100, they will start shooting. Since all that matters is the net change in opinion, it is perfectly possible to leave an inoffensive diplomacy ship in one of their systems and still gain points every cycle, provided the demand to leave isn’t too strong. You want to find a system where they have a colony and at least occasional ship traffic that isn’t their home system. Or you want your diplo ship to follow one of their exploration ships around.