If I understand your question correctly, there are two related things:
1) Improve Relations - is the diplomatic action you can undertake that improves another country's opinion of your country. It requires a diplomat to undertake it, and it creates or increments an opinion modifier called:
2) Improved Relations - this opinion modifier is probably the most important one in the game, since it's the one you can control and you can get it to +100 (+200 for subject nations). It gets augmented every time you send a diplomat on the mission, and at the beginning of every month if the diplomat stays on that mission. Like all opinion modifiers, though, it decays (at the beginning of each year).
As you've probably observed, the aggregated opinion modifiers influence whether a target nations will do things like accept an alliance, royal marriage, military access, call to arms, etc.
AE is a negative opinion modifier, which also decays. The rate of decay is influenced by your Better Relations Over Time modifier. TBH, I don't know if there is place you can see the overall bonus for BROT. In each nation's opinion mouseover, though, where you see the AE impact, you'll also see the current decay rate.