When we use the word "to buy" and when the word "to purchase"? I do not understand the difference between these words.
If it comes to the sense of obtaining something by paying money, both "purchase" and "buy" are usually used interchangeably as a verb. Nevertheless, the use of "buy" is more common in daily conversational English than that of purchase. The use of purchase is a bit formal.
In some contexts, however, it sounds more natural to use buy instead of purchase such as I will buy him the book/let me buy you a drink.
Similarly, when you use these words as a plural noun, it's more natural to use purchase in some contexts such as she opened the bag of her purchases/her purchases were so heavy that she had to take a taxi. Besides, you often say in informal English when you buy something at a low/high price "it's a good/bad buy". "It's a good/bad purchase" sounds a bit weird.
They are essentially the same. To purchase is perhaps a bit more formal.
In regular conversation with friends and colleagues, "I bought that". I'd use purchase more when writing formal reports, or talking to your boss "The company purchased 25 reams of paper."
They mean exactly the same thing 99.9% of the time, and typically you can freely use purchase where you'd use buy and vice versa.
One exception: There at least one expression (possibly regional) involving buy that would sound weird if you used purchase, for example:
A: He just told me that she's sick and won't be coming in today.
B: I don't buy that. She probably just is hungover from too much partying last night.
Here, "I don't buy that" means "I don't believe that" - using purchase here would not work.
The saying "bought it hook, line, and sinker" - meaning to have believed someone's words later discovered to be deceptive - also would not work with purchase.