1. Rewarding people according to how much they self-sacrifice, rather than how much they benefit others.
This is nothing new- it has been around since the days of humans and animals being sacrificed for the "benefit" of some greater power, typically a god or goddess. What it means is that people are rewarded primarily for making their own lives worse, rather than for helping to make others' lives better.
For example, in industry it is normal for employees to be rewarded and judged primarily according to how much of their time they set aside for work, rather than the quality, quantity and value of their output, even in jobs where, above a certain point, there is no need to have regimented "office hours". Mothers are more likely to be highly-regarded for making heavy sacrifices to their careers, interests and friendships for the sake of putting family/children first, than for putting a lot of effort into making their children happy. When we have conflicting priorities A and B, and B is more important than A, we are often more highly-regarded if we sacrifice A, than if we find an effective method of fully addressing both A and B.
Sometimes self-sacrifice can be correlated with making others' lives happier, but the emphasis is wrong- the healthiest approach is to emphasise rewarding people for the extent of their positive contributions towards others.
2. Aiming for equality of outcome rather than equality of opportunity
Yes, it can be annoying when certain people seem to get more than they deserve, but I would argue that, if it isn't at anyone else's expense, it's not necessarily a bad thing, for although it results in inequalities, it adds to the collective well-being of society. It is people getting less than they deserve that is by far the bigger problem.
It is hard, and sometimes impossible, to achieve equality of outcome without bringing everybody down to the lowest common denominator ("If some can't have it, why should anybody?"). In particular, different people have different strengths and weaknesses, so it is not possible to make everybody a "winner" in any one subject area. The most effective way to get as near as we can to a society of "winners" is rather to help give everybody the opportunity to make the most of their individual strengths.
One example of such a failed "equality of outcome" policy was New Labour's drive in the UK to get everybody into university, which has resulted in a significant devaluing of university degrees (as university used to be primarily the domain of people who are strong at academic subjects, and we had to introduce vocational-oriented degrees to enable the less academically proficient to get into university). The government has then struggled to find funds for the spiralling numbers of university students, resulting in tuition fees rising to £9,000 per year. Far from making it "fair", this policy has ended up grossly unfair on the people who struggle with being burdened with large debts, and the strong academics who were always destined to get into university but now have to deal with much larger student debts and a devaluing of their degrees because of a failed government policy.
3. "Nice guys finish last"
As I hope I've established above, a "fair" society (or the nearest we can get) should have winners and losers- in an ideal world, we should all get the opportunity to make the most of our strengths, with the winners being those who make best use of their strengths, and the losers being those who don't bother.
But in the real world, the "winners" are often the people who know how to bend rules to their advantage and bully others into submission without infringing on any higher authorities' rules, while the "losers" are the ones that they trample on.
I've often read that in relationships, "nice guys finish last" in the sense that women tend to want "nice guys" as platonic friends rather than as boyfriends, which makes it harder for "nice guys" to find partners. I think there may be some truth in this, but I associate the concept more with the sort of "nice guy" who respects women's desire to be friends with him, and then pays for it when a gang of bullies destroy his reputation by circulating rumours that he secretly wants to have sex with them and the masses believe the rumours.
This is something that needs addressing because it encourages people to try to become "winners" by bullying people rather than being nice to them.
Political implications
I suggest that many "socialist" ideologies fall down because they encourage a significant element of point 2, while many "capitalist" ideologies allow the problem outlined in point 3 to persist without opposition, while point 1 is a recurring problem across most political ideologies. Food for thought perhaps.