The supporting words 'the' and 'a' point to the difference between a definite and indefinite singular noun. A noun that is definite is associated with a particular object or person. 'The"'vase means, for instance, that the speaker is referring to one specific vase out of all the vases in the universe. 'A' vase means any vase out of a number of vases and can be used to refer to the general concept of "vaseness": you could, for example, say that "a vase is usually made of porcelain," denoting the category of vases rather than one particular vase.
We can expand our definition outward to encompass more definite articles, such as this, that, and those. These words concretize and specify a particular thing: 'that' woman, like 'the' woman, is not any woman, but a specific person. 'Those' teenagers are particular group of teenagers. In contrast, 'a' woman is much more hazy and indefinite. Saying I saw 'a' woman could mean any woman; 'the' woman, in contrast, always has some sort of known referent behind it, such as 'the' woman who said she would call, who stole the vase, or who saved my cat. There is no indefinite article in the plural, so the plural noun alone denotes the quality of being indefinite: for example, "teenagers milled around" is indefinite, while "those teenagers milled around" is definite.
This concept can be confusing and seem superfluous to non-native speakers whose language of origin does not include articles, but articles are integral to spoken and written English.