The simple answer is that sugar is one type of carbohydrate. Carbohydrates are a whole family and sugar is the evil step brother. Just kidding! Kind of. There is no technical definition of sugar, but for the purpose of this article it will be defined as that granulated white (or brown) stuff used in baking and poured on your cereal or coffee. Food manufacturers also add sugar to beverages and foods such as soda, juice drinks, cookies, cakes, candy, etc. Sometimes this added sugar is listed under names such as corn syrup, high fructose corn syrup, and corn syrup solids. The scientific name for white sugar is sucrose.
Carbohydrates are the broader category which includes all sweets, grains, fruit, milk, and even vegetables. Carbohydrates found in fruit are fructose; carbs in milk are lactose; and carbs in grain foods are glucose. When your body digests carbohydrates they break down into glucose when they enter your blood stream. All types of carbohydrates are broken down into glucose in your body. A strawberry will turn to glucose just the same as a piece of bread or a tablespoon full of sugar will turn into glucose. The difference is that the strawberry contains vitamin C, fiber, and approximately 80 calories for 1 cup. On the other hand, 1 cup of white sugar (sucrose) contains zero vitamins and fiber, and has a whopping 770 calories. Of course, most of you are not going to eat a cup of sugar, but you may drink a soda which contains 9 teaspoons of sugar.
If you ever turn a food package around and look at the label (which I hope you do!) you may notice that it shows carbohydrates, dietary fiber and sugars. Basically, the more fiber it has the healthier it is, and the less sugar it has the healthier it is. Fiber is also a carbohydrate, but it is indigestible and it moves through your body helping to lower your cholesterol and promote regularity. Fiber can also help stabilize your blood sugar.
Overall, carbohydrates provide fuel for your body and should not be avoided. There are both healthy carbs (whole grains, fruit) and unhealthy carbs (sugar), and with a bit of balance and moderation you can happily consume both types :).