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Carbonic acid, which is a weak acid, forms when rainwater and carbon dioxide in the air react with one another. When carbonic acid comes into contact with limestone or calcium carbonate, it can cause it to dissolve over a long period of time. In nature, this reaction leads to the formation of caves.
What we call carbonic acid is actually CO2 dissolved in water in equilibrium with H+ and HCO3^-. Carbonic acid is a weak acid, and so you might think that there would be nonionized H2CO3 molecules. That simply isn't the case.