The feeling of guilt in a person's conscience is the proper sentiment after he commits a sin against God or neighbor. It leads him to seek forgiveness and to make amends for his offense.
When a person violates his conscience by deliberately doing something he knows to be morally wrong then he ought to feel guilt or shame. Pope Francis calls this proper sense of shame a "great sentiment" and a "grace of God" that "opens the door to healing." Guilt leads a sinner to contrition, to be sorrowful for his sins. It then leads him to seek forgiveness and to make amends for his transgressions.
The proper feeling of guilt presupposes a well-formed conscience that knows right from wrong. It also presupposes a person's awareness of his actions when violating his conscience. Many in the Catholic Church and in secular society today do their best to blur a person's moral responsibility in order to shield him from ever feeling guilty. Without guilt, however, a person wouldn't be moved to reconcile with his God or his neighbor after he violated his relationship with them.
A person, who feels little or no shame when sinning, tends to be what's called psychopathic. Oftentimes, this is owing to a lack of moral formation. A person, who scrupulously feels guilt over non-sinful actions, tends to be what's called neurotic. Oftentimes their excessive guilt is fueled by a misguided moral formation that sees sin everywhere. Repeated sins tend to numb or deaden a person's conscience and diminish his authentic feelings of guilt over time.
A delicate or well-formed conscience, on the other hand, is one that is easily pricked by a sense of guilt when slightly violating God's commandments. The conscience immediately moves the person to seek forgiveness and reconciliation. After doing so, the conscience again feels at peace.
Pope Pius XII said in 1946 that the loss of the sense of sin or feelings of guilt at having sinned was the greatest sin of modern time.
See how God bestows His mercy on sinners in season two of Church Militant's Premium show, Majesty of the Faith—Confession.