These may mean slightly different things, depending what country you are in. In the US, the difference between a grocery store and a supermarket is size, and maybe location. Supermarkets tend to be large stores in shopping plazas, whereas grocery stores are smaller and tend to be closer to residential areas. Supermarkets can sell a lot more variety of things, but both sell mostly food. Department Stores, in contrast, typically don't sell much food, are in larger shopping plazas, and can sell a variety of items (clothes and electronics and sporting goods) all in the same store, or sell only one type of item (clothes or electronics or sporting goods) and not much else. I hope it helps.

In modern American usage, there is almost no difference between a grocery store and a supermarket. In some regions of the USA, one term is more common than the other. Both are usually used to describe what is actually a supermarket - a large store that sells all types of food, toiletries, and cleaning products. There are smaller grocery stores that sell mostly produce and such, but they are becoming rather rare in the USA. Usually if somebody says that they are going to the grocery store, they mean the supermarket. I do not know if this is different in other English-speaking countries. A department store is a large store that sells many different types of goods (has many departments). This almost always includes clothing, electronics, and home appliances, and can also include things like furniture, hardware, office supplies, and food (that is less common).

That was true until about 10 years ago in the USA. Now Wal-Mart and Target, the two largest department store chains in the US, have been converting many of their stores to be about 30% supermarket, including fresh produce.

department store never sells vegetables or fruit.

A grocery store sells fruits and vegetables, a supermarket sells %90 food,(other %10 is clothes and the other things) and a department store has all of them (sells everything you can think).