Meaning: This means that you own a cat. It's like saying, "There is a cat that belongs to me."
Choice of Words: The word "have" is a verb that means to possess or own something. "A" is an indefinite article that refers to one unspecified item. "Cat" is a noun that refers to a small, furry animal that is often kept as a pet. When you put them together, "I have a cat" means that you possess or own one unspecified cat.