Use "the" when you use a common noun to refer to a specific thing.
A common noun all by itself doesn't refer to any specific animal. The word "cat," all by itself, doesn't refer to any specific cat. So you need some other word to clarify which cat you're talking about.
Suppose you want to refer to a single cat. If you don't know its unique identity, or if you don't care to say which unique cat it was, you can say "a cat" to indicate a single, unidentified cat. You can say "A cat jumped off the table" in order to refer to a cat without specifying which cat it was.
If it's important to refer to a specific cat, you have to do two things. First, introduce the cat into the conversation. "A cat jumped off the table" tells us that there was a cat. Once you've introduced the cat, you can refer to it by saying "the cat."
"A cat jumped off the table. The cat went to the refrigerator and meowed." In the first sentence, we introduce some previously unspecified cat. In the second sentence, the word "the" says that we're referring to the specific cat that we have previously identified as a main topic of conversation. In this case, "the cat" refers specifically to the cat that jumped off the table.
"A" is called the indefinite article. Indefinite means that we have not yet defined which cat we're talking about.
"The" is the definite article. That means that we have defined which cat we're talking about.
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