We are in Maseches Avos, chapter five, mishnah eleven (in some versions it is numbered as mishnah fourteen). The mishnah lists "arba middos b'de'os" - four types of people according to their temperament, with regard to anger and appeasement.
"Noach lich'os v'noach liratzos - yatza s'charo b'hefsedo":
There are people who are quick to anger and become angry easily, but on the other hand are easily appeased and calm down quickly. There is certainly something praiseworthy in a person who calms down quickly and is easy to appease when someone has provoked him, but in the end the flaw of quick anger is far more serious than the virtue of quick appeasement. Therefore all the reward and benefit he gains from being easily appeased is cancelled out by the loss of being quick to anger.
The reason for this: at the moment of anger a person stumbles into many transgressions. Chazal said: "lo tirtach v'lo techti" (Berachos 29) - do not become enraged and you will not sin, and corresponding to this it says in Mishlei: "uva'al chemah rav pesha" - one who is constantly angry is full of sin. In his moment of anger a person does things he is forbidden to do and says things that should not be said, and for this reason the harm of easy anger outweighs, and outweighs by far, the advantage of being easily appeased.
It is worth noting that there is a reversed version in the wording of the mishnah: "yatza hefsedo bischaro." However, the Bartenura holds that the first version is the primary one, and we will follow that approach.
"Kasheh lich'os v'kasheh liratzos":
This person is hard to anger, but once he becomes angry he is hard to calm down and appease. There is a virtue in his being hard to anger, and a drawback in his being hard to appease, but "yatza hefsedo bischaro": in the overall balance, the advantages of being hard to anger and not getting angry often outweigh the drawback of being difficult to appease. In the bottom line, this trait is an advantage.
"Kasheh lich'os v'noach liratzos - chassid":
This person is remarkable in his character traits, since he does not become angry easily. This does not mean that he never becomes angry - every person gets angry at times, and at times it is even appropriate to be angry about certain things - but rather that he is not quick to anger, and when he does become angry, he is easily appeased. Such a person goes above and beyond, and therefore he is called a chassid.
There is a discussion among the commentators as to whether a person's tendency toward anger and the corresponding tendency to be appeased are something a person is born with, or whether they are acquired and developed through his own choices. The reasonable understanding is that both factors are at work here. In any case, this person has certainly worked on himself: for by nature, one who is hard to anger is also hard to appease, and one who is quick to anger is also quick to be appeased. He has taken the good of both sides, and therefore he is a chassid.
"Noach lich'os v'kasheh liratzos - rasha":
This person becomes angry easily, and afterward it is very difficult to appease him. He has worked on himself in the opposite direction, and has trained himself not to be among those who are slow to anger. In effect he is addicted to anger, and when he becomes angry he holds on to the grudge in his heart, and therefore the mishnah calls him a rasha.
The Version of the Midrash Shmuel:
The Midrash Shmuel has a different version of this Mishnah, and this version seems preferable. According to his approach, "lirtzos" does not mean to be appeased and pacified, but rather to appease someone else. Technically, "lirtzos" means to become appeased, but the intent of the Mishnah is appeasing others and asking them for forgiveness. Accordingly, the Mishnah lists four temperaments among people:
"Noach lich'os v'noach lirtzos": He becomes angry easily, but he is quick to apologize and it is easy to get him to appease his fellow. It is certainly a virtue to apologize quickly, but it would have been better had he not become angry in the first place, and therefore the entire advantage of quickly asking forgiveness and apologizing is balanced against the drawback of being quick to anger.
"Kasheh lich'os v'kasheh lirtzos": It is hard to anger him in the first place, but once he is angry it is very difficult to get him to appease his fellow, since he does not like to appease. Although this is a bad trait, it is a wonderful virtue that he does not become angry easily, and in the overall balance this is the more important point.
"Kasheh lich'os v'noach lirtzos": It is hard to anger him, and when he does become angry he is quick to apologize and appease his fellow. Such a person is a chassid.
"Noach lich'os v'kasheh lirtzos": He is easily angered, and when he is angry he never apologizes but always holds a grudge, and it is difficult to bring him to apologize. This person is a rasha.
In summary: The Mishnah lists four types of temperament, and in each one the measure of anger is weighed against the measure of appeasement. We learned that being quick to anger outweighs, to one's detriment, being quick to appease, for "do not become enraged and you will not sin" and "a man of wrath abounds in transgression," while being slow to anger outweighs, in one's favor, even being slow to be appeased. One who has taken the good of both sides - slow to anger and quick to appease - is called a chassid, and one who has combined both flaws - quick to anger and slow to appease - is called a rasha. And we saw the version of the Midrash Shmuel, according to which "lirtzos" means appeasing another person and asking him for forgiveness.