
der Zucker
sugar
A sweet-tasting, crystalline substance (sucrose) obtained from plants such as sugar beets or sugar cane and used as food.
Ich trinke meinen Kaffee ohne Zucker.
I drink my coffee without sugar.
sugar
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sugar
A sweet-tasting, crystalline substance (sucrose) obtained from plants such as sugar beets or sugar cane and used as food.
Ich trinke meinen Kaffee ohne Zucker.
I drink my coffee without sugar.

blood sugar
In a medical context, the short form for blood sugar, meaning the concentration of glucose in the blood.
Nach dem Essen hat der Arzt meinen Zucker gemessen.
After the meal, the doctor measured my blood sugar.
In its main meaning, 'Zucker' (sugar) is an uncountable noun (mass noun). It is therefore usually used without an article or in the singular when talking about the substance in general.
Für diesen Kuchen braucht man Mehl, Eier und Zucker.
For this cake, you need flour, eggs, and sugar.
You can tell from the context whether household sugar or blood sugar is meant. In a medical context or in combination with verbs like 'messen' (to measure) and adjectives like 'hoch' (high) or 'niedrig' (low), it almost always means blood sugar.
Mein Arzt sagt, ich muss auf meinen Zucker achten.
My doctor says I have to watch my blood sugar.
When referring to a specific amount of sugar, quantity expressions are often used, such as 'ein Löffel Zucker' (a spoonful of sugar), 'ein Stück Zucker' (a sugar cube), or 'eine Prise Zucker' (a pinch of sugar).
Möchtest du ein Stück Zucker in deinen Espresso?
Would you like a sugar cube in your espresso?
In a medical context, 'Zucker' (blood sugar) is often used with the possessive pronoun. Typical phrases are 'seinen Zucker messen' (to measure one's blood sugar), 'hohen/niedrigen Zucker haben' (to have high/low blood sugar), or 'ein Problem mit dem Zucker haben' (to have a problem with one's blood sugar).
Viele Diabetiker müssen mehrmals täglich ihren Zucker messen.
Many diabetics have to measure their blood sugar several times a day.
carbohydrate
Zucker is a specific sweet substance, while Kohlenhydrate form a broader group of nutrients. Sugar therefore belongs to carbohydrates, but not every carbohydrate is simply sugar. Use Zucker when the sweet substance is meant, and Kohlenhydrate when referring to the nutrient group.

Das Müsli enthält viel Zucker und schmeckt deshalb sehr süß.
The muesli contains a lot of sugar and therefore tastes very sweet.
Das Müsli enthält viele Kohlenhydrate und liefert dem Körper Energie.
The muesli contains a lot of carbohydrates and provides the body with energy.