In psychology, projection is the act of assuming that other people see the world in the same way that you see the world.
An example of projection would be a Christian who assumes that everyone should see God in a sunset or recognize the authority of the Bible. Since it seems so natural and obvious to the Christian, she projects these feelings onto the rest of humanity.
Some of the most interesting cases of psychological projection occur cross-gender. This is particularly true in a society that strives to be gender neutral.
For men, physical beauty is the most desirable trait in a female. So an act of projection would be to assume that for women, physical attractiveness is the most desirable trait. While physical attractiveness is important to women, it is far less important to women than it is to men. So the man who puts all of his effort into his appearance is not getting the best bang for his attraction buck. But the reason he puts all this effort into his physical appearance is because of the strong illusion that's caused by psychological projection.
The same goes for sensitivity and vulnerability. Men desire gentle, sensitive, vulnerable, submissive female mates. But many men project these desirable traits and assume that women want them in men too. In fact, most women prefer men who are strong, confident and inviolable. And they perceive sensitivity as a weakness.
Finally, men love reason and truth. It is a biologically ingrained trait that most likely developed as a strategy for tracking, hunting and killing fat and protein rich sources of meat. The female brain did not have the same selection pressures and thus their brain is not as weighted towards logic and truth (this manifests itself in the fact that many women feel threatened by debate, whereas men find debate to be one of the highest pleasures: the brain rewarding them for an evolutionarily important skill). While it is true that women can be attracted to intelligence in men where it is manifested in external skills and debate with other men, most women are not attracted to the direct expression of intellect within a relationship in the form of cold, hard facts. Rather, they are more attracted to things like cocky humor. Attraction is about feelings: for women, truth and logic create cold feelings, sarcasm and cocky humor are hot.
Let's reverse the genders now. For most women, the most desirable trait in men is high social value. High social value in men is achieved in many different ways: careers, unique skills, wealth, success, etc. Many women are attracted to older men precisely for this reason: because older men have had more time to gain high social value and, because of their life experiences and station in life, tend to present themselves as being more stable, inviolable, confident and having more access to valuable resources.
As we noted earlier for men, biologically, beauty is the trump card in attraction. The animal hind-brain of a man takes over in the presence of beauty and despite any alternative principles, he is largely rendered incapable of discretion. Despite this fact, many women project onto men what they themselves value (high social value, etc.). So women will often seek to impress men by developing their social value in the form of skill acquisition (learning an instrument), career advancement, and development of intellect. Projection in women tends to hit a peak as youthful beauty starts to decline. It is around this time that many women will seek to earn a new college degree and move up the career ladder.
These are just a few examples of projection between the genders as it relates to attraction. However, there are hundreds of other cases ranging from expectations for how time is spent, to responsibilities around the house, to expressions of love (one person might feel like he is expressing love, but the expression of love is not being perceived because the other person perceives love in different forms than the way it is being expressed).