There are many reasons why we, as humans, suffer in this world. Much of it is due to the fact that we're in the world. Remember, due to the sin of Adam, creation was subjected to death and decay; we have bodies which get sick, grow old and die.1 The laws of nature, established by God, affect much of what we experience.
Through the ages, disease, chemicals, and many other things have caused many physical problems. Our genetics have been altered, and in many cases, substances which were designed to help us, did just the opposite. Thalidomide, which caused babies to be born without legs or arms, is a good example.
Many bad things also happen to us because of wrong choices on our part. An example is one who steps in front of a car and is killed. Another example is one who drinks alcohol to such a degree that they end up with liver damage.
Satan, the arch enemy of God, also has a lot to do with suffering. He delights in hurting mankind, and in trying to get us to turn our backs on God.
Another reason we experience suffering in this world, is that God allows hardship, at times, to come our way to produce perseverance and character in our lives. As the Bible says: "Consider it pure joy, my brothers, whenever you face trials of many kinds, because you know that the testing of your faith develops perseverance. Perseverance must finish its work so that you may be mature and complete, not lacking anything."2 We, as humans, grow through adversity; when our faith is put to the test. As it has been said, "In this life we go from one catastrophe to another, but they have less and less effect on us as we allow God to bring us through them." God prepares us for the future.
1 (Genesis 3:17-19)
2 (James 1:2-4)