If you are trying to get something out of them, their attitude matters. If they are your KeyCiv, tech gift them everything you have (I usually only hold back one "vital" tech) to get your beaker costs down. This will usually drive them up to Worshipful, whereupon you can exchange maps (VERY useful) and ask for a gift in return (tech or gold).
If you are not gifting an ally occasionally, over time their attitude will decline. If you stab someone in the back by breaking an alliance or peace, their attitude will go WAY down and stay there for a very long time. Other civs also take notice of backstabbers and sometimes their attitudes go down too.
When you play a game like One City Challenge or Early Landing you often deliberately decline on the Power Rating, so getting the other civs attitudes up so you can trade maps and ask for handouts is critical. If you just play to be Supreme, there is a built-in bias in the game that other civs will band together against you.
When they go from Neutral to Uncooperative it seems to indicate they are on the verge of becoming really unhappy, usually leading to a "sneak attack". I have seen this most often with the Mongols, also with other Militaristic civs, that if you trying to stay at peace with them you need to give them something ASAP.
From Neutral there are four "better" attitudes, Receptive, Cordial, Enthusiastic and Worshipful, and four "worse" attitudes: Uncooperative, Icy, Hostile, and Enraged. When a civ attacks you but is not Hostile or Enraged they are usually pretty easy to talk into a cease-fire or peace.