It is my observation that most people are fairly good at seeking forgiveness. But penance is an entirely different thing.
To seek forgiveness, you must subdue your pride. Doing penance is voluntarily submitting to some form of unpleasantness.
"Isn't seeking and receiving forgiveness enough? After all, once I am forgiven, I'm forgiven."
When I was young I asked the same question. A priest explained forgiveness and penance like this: Suppose you are in the back yard playing baseball. You hit a home run right through the neighbor's window. Rather than run and hide, you go to the neighbor, confess what you have done, and ask his forgiveness. Although the neighbor forgives you, the window is still broken. You still have the responsibility to make it right. Making it right is the doing of penance.
Going to an intangible God that you know is merciful and loves you for forgiveness is easy. Yet even though we seek forgiveness from our fellow man and God, and receive it, harm has been done. Doing penance is where the rubber meets the road.
In the example of the broken window, the person may be able to fix it himself, pay someone else to do it, or work for the homeowner until the debt is paid. In the criminal justice system, the offender may have an obligation to make restitution to his victim as well as an obligation to perform community service to make up for offending the peace and dignity of society. How do you make up for offending God?
Typically, sin results from an unwillingness to do without. I want something but can't afford to pay for it and take it. I don't want to deal with frustration so I yell, curse or resort to violence.
Penance typically means voluntarily doing without. In the baseball example, the person does without his money, free time or both. Penance disciplines the mind and body to resist temptation. Even when the harm cannot be undone, doing penance helps the penitent avoid the sin in the future.
What are common forms of penance?
In the year 1170, the King of England, Henry II, had the Arch Bishop of Canterbury, Thomas a Becket, killed. His penance was to crawl on his knees from London to Rome. He actually did it! They don't do penance the way they used to.
Among the most common forms of penance is prayer. How strange that communication with God is a form of penance. But, among theologians, sin is likened to a spiritual illness. Penance should be calculated to cure the spiritual illness. Prayer is like a shot of spiritual antibiotic.
Perhaps the second most common form of penance is fasting and abstaining from enjoyable things. Most Americans have never truly known hunger. Try going seven days without eating anything. By the third day it is difficult to focus on anything besides your hunger.
Sometimes the penance includes undoing the harm your sin has done.
Most importantly, penance helps to cleanse the soul. It seems to me that as Jesus had to suffer for our sins, and that he did so voluntarily, doing penance is a part of redemption.