("A lot" is two words; "allot" is a verb, relating to "allotment" )

This is a complicated question.

As far as I know, all Western languages (All Indo-European...Germanic, Romance, Hellenic, etc.) use articles, indicating "the" or "a," or at least use various grammatical cases to make the nouns or noun phrases more distinct.

Asian languages, however, do not. That being said, they are still able to communicate "the" or "a." Or rather, all languages on this planet are able to stress limit/delimit nouns, phrases, etc.

So Chinese does have "the" or "a," but it is achieved through different means. And you don't necessarily have to use 这 or 那 (this or that).

One example is the Chinese topic --> comment structure. If you make something the "topic" then it is similar to adding "the."

Example.

你买了花儿没有？
I would translate as indistinct. "Did you buy flowers yet?" or "Did you buy [some] flowers yet?"

Now topic-comment.

花儿，你买了没有？
Distinct. I would translate as "Did you buy THE flowers yet?" (The flowers in question that the speaker already is aware of)

"A" is mostly extraneous. It is not necessary.

他是好人
他是个好人
他是一个好人
... All of the above would be translated as "He is a good person." Granted, if you add different stress/inflection, that can change the intended meaning. If you really stress the 一 then it does literally does become "He is [but] ONE good person." Something like that.

And as for flowers, if you wanted to be really really specific, you could always ask "那一朵花儿，你买了没有？ "Did you buy that one flower yet?"

The above answers are satisfactory, but I thought I might add a little food for thought.  (Just so you know, Chinese grammar sometimes confuses me, so what follows should be somewhat if not exactly correct.)

It seems to me that a lot of Chinese put 'that' (那个) where we might use 'the' instead.  For example:

我想买那个我昨天看到的书。
I want to buy the book that I saw yesterday.

Actually, 那个 seems to be more of a placeholder for when somebody hasn't quite figured out what he or she wants to say in a sentence.  Thus, I believe the above sentence's meaning is usually more like:

I want to buy 'that... uh...(what do I want to say?)' book that I saw yesterday.

The Chinese language does not have a word for "the." For example, "the book is big" is simply translated as "书很大."

To the above poster, there is a translation for "a." "I have a book" is "我有一本书."

Correct me if I am wrong somebody but I don't believe there is a direct translation for the or a.