There are a variety of unhealthy characteristics of sugar and some of these characteristics depend on the type of sugar that you're talking about. I'll assume you're referring to common sugars like sucrose (table sugar), dextrose (corn sugar) and high-fructose corn syrup.
Sugar can negatively impact your immune system, contribute to a loss of tissue elasticity, cause stress, increase insulin levels (which promotes rapid weight gain), help cancer spread more rapidly (cancer is fueled by glucose, so many cancer treatments also include watching your blood-glucose levels carefully), and other problems that can lead to heart disease, diabetes, tooth decay, arthritis, kidney stones, depression...you name it.
Here's a list of 146 Reasons Why Sugar is Ruining Your Health.
Obviously everything depends on moderation and it's possible to eat sugars while still maintaining a healthy diet. However, avoiding sugar-heavy foods and drinks like candy, soda pop, many cereals, etc. will certainly help you live a longer, happier, healthier life.