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The Mishnah states that someone who calls one by a nickname others have given him has no share in the world to come. What exactly does this mean? How is this defined? Can you not call your friend by a nickname? Please cite sources

The nicknames mentioned in halacha are in the context of derogatory nicknames, used to shame others.

The source is in Baba Metzia 58b: three do not ascend from the world to come and these are

[...] one who humiliates another in public; and one who calls another
a derogatory name

The Shulchan Aruch (Choshen Mishpat 228:5) codifies that one should not call his friend by a nickname, even if he is used to it, if the intent is to shame him.

The Rambam (Hilchot Tshuva 3:14) lists practices that remove one from the world to come

[...] he who gives an alias name to his friend; he who calls his friend by
his aliases; he who shames his friend publicly; [...]

but adds that one can repent from it

if he repent from his wickedness and died in a state of repentance, he
is, indeed, of the sons of the World to Come, as there is not a thing
to stand in the way of repentance

Therefore calling someone by a friendly nickname that he accepts or commonly uses does not fall within the bounds of this prohibition.

There are two issues here. One is what are the parameters of a "nickname" that is problematic, and another is what is meant by the punishment of "no share in the world to come. Regarding the first issue, there are a number of relevant Talmudic passages. In Bava Metzia 58b we find the following:

אמר ליה אביי לרב דימי במערבא במאי זהירי א"ל באחוורי אפי דאמר רבי חנינא הכל יורדין לגיהנם חוץ משלשה הכל ס"ד אלא אימא כל היורדין לגיהנם עולים חוץ משלשה שיורדין ואין עולין ואלו הן הבא על אשת איש והמלבין פני חבירו ברבים והמכנה שם רע לחבירו מכנה היינו מלבין אע"ג דדש ביה בשמיה

Abaye asked R. Dimi: What do people [most] carefully avoid in the West [sc. palestine]? — He replied: putting others to shame. For R. Hanina said: All descend into Gehenna, excepting three. 'All' — can you really think so! But say thus: All who descend into Gehenna [subsequently] reascend, excepting three, who descend but do not reascend, viz., He who commits adultery with a married woman, publicly shames his neighbour, or fastens an evil epithet [nickname] upon his neighbour. 'Fastens an epithet' — but that is putting to shame! — [It means], Even when he is accustomed to the name. (Soncino translation)

However, in Taanit 20b we have the following passage:

שאלו תלמידיו (את רבי זירא ואמרי לה) לרב אדא בר אהבה במה הארכת ימים אמר להם מימי לא הקפדתי בתוך ביתי ולא צעדתי בפני מי שגדול ממני ולא הרהרתי במבואות המטונפות ולא הלכתי ד' אמות בלא תורה ובלא תפילין ולא ישנתי בבית המדרש לא שינת קבע ולא שינת עראי ולא ששתי בתקלת חברי ולא קראתי לחבירי בהכינתו ואמרי לה בחניכתו

The disciples of R. Adda b.
Ahaba asked him: To what do you attribute your longevity? — He replied: I have never displayed
any impatience in my house, and I have never walked in front of any man greater than myself, nor
have I ever meditated [over the words of the Torah] in any dirty alleys, nor have I ever walked four
cubits without [musing over] the Torah or without [wearing] phylacteries, nor have I ever fallen
asleep in the Beth Hamidrash for any length of time or even momentarily, nor have I rejoiced at the
disgrace of my friends, nor have I ever called my neighbour by a nickname given to him by myself,
or, some say by the nickname given to him by others. (Soncino translation, my emphasis)

There is a very similar account in Megillah 27b:

שאלו תלמידיו את רבי זכאי במה הארכת ימים אמר להם מימי לא השתנתי מים בתוך ד"א של תפלה ולא כניתי שם לחבירי ולא ביטלתי קידוש היום אמא זקינה היתה לי פעם אחת מכרה כפה שבראשה והביאה לי קידוש היום

R. Zaccai was asked by his disciples: In virtue of
what have you reached such a good old age? He replied: Never in my life have I made water within
four cubits of a place where prayers have been said, nor have I given an opprobrious epithet to my
fellow, nor have I omitted [to perform] the sanctification of the [Sabbath] day. I had a grandmother
who once sold her headdress so as to bring me [wine for] the sanctification of the day. (Soncino translation, my emphasis)

These latter two passages make it seem as if not nicknaming someone is an act of special righteousness, while the first passage made it seem that nicknaming someone is one of the worst things you can possibly to.

Tosafot on the passage in Taanit explains that it's only a derogatory nickname that is problematic but a positive nickname is fine:

בהכינתו. פירוש בכינוי שמכנין אותו בן אדם לגנאי בחניכתו פי' כינוי שם משפחתו שם לווי כמו שם חניכת אבות בגיטין עד עשרה דורות כגון רבי אברהם אבן עזרא שכל בני משפחתו היו נקראים כך ודוקא לגנאי אבל לשבח מותר

Tosafot on the passage in Megillah explains that the especially praiseworthy act described is that the sage wouldn't even use a non-derogatory nickname. And the passage in Bava Metziah is referring to degrading the family name:

ולא כניתי שם לחבירי. אפילו כינוי דלא הוי גנאי והא דאמרינן דהמכנה שם לחבירו אין לו חלק לעולם הבא היינו בכינוי של פגם משפחה

Thus, according to Tosafot a nickname which degrades the family name is what forfeits your share in the world to come, a neutral nickname and a positive nickname are permissible, and certain righteous individuals refrained from even these (certainly neutral names, unclear about positive names).

Rambam in Hilchot Deiot 6:8 writes simply:

כך אמרו חכמים המלבין פני חבירו ברבים אין לו חלק לעולם הבא לפיכך צריך אדם להזהר שלא לבייש חבירו ברבים בין קטן בין גדול ולא יקרא לו בשם שהוא בוש ממנו ולא יספר לפניו דבר שהוא בוש ממנו

Our Sages said: "A person who embarrasses a colleague in public does not have a share in the world to come." Therefore, a person should be careful not to embarrass a colleague - whether of great or lesser stature - in public, and not to call him a name which embarrasses him or to relate a matter that brings him shame in his presence. (Touger translation)

In Hilchot Teshuva 3:14 he writes:

ויש עבירות קלות מאלו ואעפ"כ אמרו חכמים שהרגיל בהן אין להם חלק לעוה"ב וכדי הן להתרחק מהן ולהזהר בהן ואלו הן המכנה שם לחבירו והקורא לחבירו בכינויו

There are other sins which are less severe than those mentioned. Nevertheless, our Sages said that a person who frequently commits them will not receive a portion in the world to come and [counseled] that these [sins] be avoided and care be taken in regard to them. They are:

one who invents a [disparaging] nickname for a colleague; one who calls a colleague by a [disparaging] nickname; (Touger translation)

R. Yaakov Ben Asher formulates the law as follows, in Tur C.M. 228:

ויזהר אדם מלכנות שם לחבירו כדי לביישו אפילו שהוא רגיל באותו כינוי כיון שכוונתו לביישו

A man should be careful of nicknaming his fellow in order to embarrass him, even if [the fellow] is accustomed to this nickname, since his intent is to embarrass him.

This is repeated by R. Yosef Karo in Shulchan Aruch C.M. 228:5, R. Mordechai Yaffe in Levush C.M. 228:5, and R. Yechiel Michel Epstein in Aruch Hashulchan C.M. 228:2.

R. Avraham Azulai in his glosses to the Levush writes, based on Tosafot above, that degrading the family name forfeits your share in the world to come (whereas apparently a general derogatory nickname is only forbidden but does not forfeit your share in the world to come):

'ובשם של פגם משפחה אין לו חלק לעולם הבא תוס' מגילה כ"ז ב

Regarding the second issue, of what it means to forfeit your share in the world to come, in this answer I cited a couple of sources arguing that such statements are hyperbolic. Additionally, Rambam in his Commentary to the Mishnah (Sanhedrin 10:1) writes as follows:

וכבר הזכירו דברים זולת אלה שהעושה אותם אין לו חלק אמרו המלבין פני חבירו ברבים אין חלק לעולם הבא והקורא לחברו בכנויו והמתכבד בקלון חברו לפי שלא יבוא מעשה מן המעשים האלו ואף על פי שהם קלים לפי מחשבת החושב כי אם מנפש חסרה שלא הגיעה לשלמות ואינה ראויה לחיי העולם הבא (Kafih translation)

And they have already mentioned other things besides these that one who does them does not have a share. They said that one who whitens the face of his friend in public has no share in the world to come, and one who calls his friend by his nickname and one who glorifies in the disgrace of his friend, because though these actions are insignificant in the mind of the thinker they only come from a lacking soul that has not reached perfection and is not fitting for the life of the world to come.

Thus, according to Rambam, it may be that this act itself does not cause one to lose his share of the world to come; rather, such an act is indicative of the type of person who would anyway not have a share in the world to come.