I have seen example sentences, but they didn't use a (the indefinite article). Does the equivalent of a exist?
Esperanto doesn't have the indefinite article, but only the definite article la. If you want to say (for example) a book, you just say libro, while la libro means the book.
See also The Sixteen Rules of Esperanto Grammar, Articles for more details.
The word "a" is the indefinite article in English. However, an indefinite article doesn't exist at all in Esperanto. http://lernu.net/en/gramatiko
You have to figure out what sort of meaning are you trying to convey with the article. Either it is unimportant and so you leave it out, or you have to convey that meaning differently. Often an indefinite article just marks the absence of a definite article. For the cases where the indefinite article has some extra meaning you could use iu (when by a cat you really mean some cat) or perhaps unu (when by a cat you really mean one cat, emphasizing the number), or sometimes even a different word completely.
Note that in some languages like French the indefinite article is in fact the same word for one anyway. In Czech and other slavic languages we have a "worse" problem: We have no articles whatsoever, so when a meaning normally conveyed by an article is required you have to resort to these extra words. But, I'd bet vast majority of cases when articles are used, they are really extraneous as far as meaning/understanding and are required purely by grammar and/or style (so of course you shouldn't leave them out).