The following discussion is my attempt at a response to this question. However, there are numerous books, articles, and websites that address this question more fully. One classic on this subject is C. S. Lewis's book, The Problem of Pain.
It is true that the problem of evil and suffering is a significant problem that is beyond our human ability to definitively answer. The problem is often stated as follows: "Evil is inconsistent with the notion of a loving, all-powerful God. Either God is not all good, or God is not all-powerful." However, those who pose this dilemma assume that suffering is meaningless. However, we can escape this dilemma by maintaining that suffering has a purpose that is beyond our limited understanding.
Of course, evil and suffering are only problems within Judeo-Christian theism, which presents an infinite, yet personal God. If you are a Deist and maintain that the Supreme Being is the "proverbial watchmaker" who is indifferent to the struggles of the human race, you have a different problem---why call evil, evil? On that view, pain, suffering, and injustice are normal and just the way things are. Or, if you are an eastern monist, suffering is part of the process of passing through multiple reincarnations or the outcome of an individual's behavior as dictated by the impersonal law of karma.
The Christian faith offers an interrelated set of understandings that, while not providing a final comprehensive answer, do offer some coherent approaches that move us towards greater clarity. Complete clarity and resolution must wait until the end of history when as the Apostle Paul states, "we will know as we are known". Our ability to comprehend the reasons things happen in this world as they do is limited. Paul, stated in his first letter to the Christians at the city of Corinth (I Cor. 13:12) that our perception or understanding of reality in this life is like the blurred image a person saw in a mirror of the first century era. Christians regard the character of God revealed in Christ as the basis for their trust in the wisdom and justice of God. We believe that God has good and sufficient reasons for permitting suffering that we cannot understand in this life.
Some people ask, "Why doesn't God simply remove all evil from the world?" "Why does God allow humans to suffer?" Perhaps we should first think about the major sources of human suffering. Of course, accidents and natural disasters do occur, because we live in a fallen world, which is not in accord with God's original purpose for creation. Consequently there are earthquakes, hurricanes, and disease. However, it seems that a large part of human suffering is caused by human beings themselves. Suffering is often the direct or indirect result of human actions that are motivated by greed, cruelty, or carelessness. Within the Christian understanding, humans are viewed as "free moral agents". In other words, we have the ability to make moral choices. For some reason, God has allowed for the possibility of human choices that contradict His will. It seems that this possibility is necessary for human freedom to exist at all. I suppose God could have created a universe in which all intelligent beings conformed to His will perfectly. However, those intelligent beings would be something like automatons. It follows then that if God were to obliterate evil from the universe, God would have to eliminate true human freedom as it currently exists. One could ask why God could not have created a world in which intelligent beings freely chose to do God's will. Apparently, for some reason beyond our ability to know, God decided to allow for the possibility of evil.
The most profound teaching of the Christian faith is the Incarnation. God entered into the human condition in the person of Jesus of Nazareth and experienced all the struggles and pains of life. Finally, at the instigation of the Jewish authorities, He was sentenced to death, tortured, and crucified by the occupying Roman government. Crucifixion was one of the most painful, cruel deaths a person could experience. However, Christians believe that Jesus's death and resurrection accomplished God's victory over sin and death. Likewise all who put their faith in Christ have the assurance that their suffering is not in vain and will be satisfied in ultimate justice and victory.
God often provides comfort and healing for our traumas and suffering in this life through a variety of secondary means, such as family, friends, medical interventions, wise counsel, the passing of time, and sometimes even through miracle. In addition, Christians have the certainty that beyond this life all hurt and trauma will be left behind as we come into God's presence either through death or when Christ returns at the end of history. Those who could not obtain justice in this life, will see justice done in a final judgement.