Chazal compare someone who gets angry to an idol worshiper – because they lose control and no longer can think rationally. That sort of anger is never good or healthy.
Yet anger of a different sort can be good, sometimes, I think. Say someone gets angry at an injustice, and it propels him to take action to correct that injustice. But that is not the type of anger where one loses control, I am not sure it would even qualify as the basic definition of anger.
Slow to anger means it takes a lot to get angry, the person does not easily get angry.
read ibn Ezra on Kohellet, he explains that there is a time and place for every emotion. Anger towards an idiot is beneficial since it tells him where to draw the line. However a person should not have pent up anger in himself.
The best is NOT to take my word, but read ibn Ezra, he is terrific (a slight bit difficult, but worth it).
I think anger can refer to either passion, rage or contempt. Contempt should be reserved for the truly awful people of the world, and rage is generally destructive, but can have good results when it happens to run into something that has to be destroyed.
I believe that that Rambam (or another classic source) says that it is forbidden to be angry at all, but one can pretend to be angry if needed, for example to properly discipline children.
From Mishlei to modern rabbonim, there are countless Torah teachings on how important it is to avoid getting angry, and how to do so. We should all review them on a regular basis and work on ourselves to eradicate this evil trait.