Usage of an articles "a" and "the".....

A (or 'an') = indefinite article

The = definite article 

A hat = any hat, it doesn't matter which one. It can get quite specific, without referring to exactly one hat.

A small green hat with feathers could be lime green with ostrich feathers or moss green with chicken feathers. 

The hat = this specific hat I am wearing on my head. (And happens to be emerald green with peacock feathers.)

Also,  see this recent GA post on the definite article.

http://answers.grammarly.com/questions/1618-grammar-rule-for-using-an-article-with-the-verb-form-use/#answer_1768