"Penitence" isn't something we discuss much these days. To clear up any confusion with similar words: penitence is an attitude of humility about ourselves, an awareness of our own struggles and imperfections, and a renewed dedication to reject any harmful or unkind thoughts, attitudes, and practices by cultivating the loving and holy things that are born of God's word.
So much of that is foreign to our culture. Then again, we're called to live as foreigners and strangers in this world, living not by the standards and practices of the passing secular fads, but as citizens of the kingdom of God, and ambassadors of the kingdom of God. So rather than dial back, let's press on: the first foreign concept is humility.
Humility is not an unhealthy self-loathing that uses our flaws as a pretext for self-hatred or self-abuse, but instead a healthy self-awareness of our own imperfections, leading to a compassion towards both ourselves and others. We reject the idea of being "better than" other people even within our own minds. We reject the idea of being "better than" other people without rejecting the idea that some actions are truly better than others. While modern seculars see the opposite of "judgmental" as "tolerant" -- anything goes! -- I'd suggest that the opposite of "judgmental" is "humble": not anything goes, but the line between right and wrong actions, right and wrong attitudes, is not used to divide people from each other, not used as a tool of control, belittlement, harassment, ostracism, or exclusion. Instead, the honest recognition of our faults makes us brothers and sisters with each other, every last one. And it does this without losing the idea that there is such a thing as good. The problem with being judgmental is not placed on the ideas of right and wrong, but on the attitude of arrogance.
In the almost-foreign idea of penitence, we have to stretch beyond our modernist shrinking spiritual world to reclaim another almost-foreign idea: sin. Remember that in embracing humility we have already rejected the practice of spiritual arrogance. Someone who does not treasure humility cannot approach the concept of sin without doing grave damage to themselves or others. (I suspect that's a large part of why the very idea -- that there is such a thing as "sin" -- is shouted down fiercely in some circles, or redefined as something that other groups do. Anything to protect ourselves from that self-awareness of our own flaws.) The concept of sin includes some thoughts that are at odds with the prevailing winds of our times: that there really are legitimate moral standards, ones which are true and right -- and universal. It's not possible to have real repentance without a concept of legitimate moral standards.
Some of the traditional practices of Lent are beginners' exercises in humility and self-control, such as modest levels of fasting and prayer. A short fast can awaken us to how much we can be controlled by even simple and healthy appetites, and how little harm is done by taking a break from self-indulgence as we tell ourselves "no" in an area where it's typical for our society to have little self-control.
When it comes to our own faults, penitence opens our eyes. When it comes to others' faults, our self-awareness in turn opens our hearts. Penitence is a fellowship-builder. Penitence is a door-opener.