Most people have a good sense of what is fair and what is not fair in any concrete situation they are involved in. It's not fair to discriminate against qualified people because of their race or gender or sexual orientation when deciding who is hired for a job or admitted to a school; it's not fair to discriminate against a race or gender or sexual orientation in deciding who is allowed to patronize a store or hotel or restaurant or how well a person should be treated in a hospital. People who support the egalitarian values of no-rich-and-no-poor equality and mutual aid, when commenting on the same specific situation in which somebody is treated in a certain way, are likely to agree about whether that person was treated fairly or not.
The formal dictionary definition of fair, in the sense we're interested in here, however, includes several slightly different meanings, some of which are consistent with how most egalitarians use the word, but others of which are not necessarily so depending on the context.
One dictionary's definition for the meaning of fair, for example, is:
Here's the problem with the dictionary definition. It does not question the rightness or wrongness of the "established rules"; it says something is fair if it conforms with these established rules. But egalitarians don't think the currently established rules of our society--rules that maintain class inequality--are right or fair, and therefore they wouldn't say that if somebody is treated in a manner "conforming with the established rules" then they are necessarily treated fairly. Justice in our society based on class inequality--as discussed here--is, in other words, not just, even if people are treated in conformity with the established rules in a manner "marked by impartiality and honesty."
For egalitarians, fairness means being treated in a manner "marked by impartiality and honesty, free from self-interest, prejudice, or favoritism, conforming with the established rules* WHEN THE ESTABLISHED RULES THEMSELVES CONFORM TO EGALITARIAN VALUES.
The dictionary definition quoted above ends with "consonant with merit or importance" as in "due a fair share." In our society based on class inequality, people are paid different wages or salaries according (officially, at least) to their different merit or importance and this is considered by anti-egalitarians to provide people their "fair share" of social wealth.
But egalitarians don't think that this provides people their "fair share" of social wealth. Egalitarians think that the principle for determining how much wealth is a person's fair share is this:
"From each according to reasonable ability, to each according to need or reasonable desire with scarce things equitably rationed according to need."
Fairness in the egalitarian sense of the word is a key part of the egalitarian values of equality and mutual aid, and it is the basis for much of this proposed Egalitarian Bill of Rights, which begins by stating that all human beings have the same rights.
* Sometimes established rules are expressed explicitly, perhaps in writing, but sometimes they are only expressed implicitly, for example by a long-standing custom that implies a rule. For example, there is a long-standing custom in many places that in swimming competitions women compete only against women, because it would be as unfair to make women compete against superior-strength (specifically due to having gone through male puberty) men as it would be to make women compete against dolphins. The rule that women only compete against other women in swim matches is consistent with egalitarianism. Therefore, because women spend many years training hard to excel in competitive swimming on the assumption that they will be competing only against other women, it is unfair to then change the rules of the game and make them compete against men.