The definite article, the, is used to introduce something that can be identified uniquely within the context of the utterance or of general knowledge. For that reason, the is typically used for "old" information. If I say "bring the chair," I assume you already know which chair I'm talking about.
The indefinite article, a/an is used for situations were the reference is not uniquely identifiable. If I say "bring a chair", I don't have any particular chair in mind.
The demonstratives are this, that, these, and those. Like definite articles, they refer to old information. But they also point to specific things: this book or those children. That "pointing" establishes a relative spatial relationship, which is reflected in the contrast between this/these, used for items that are close to the speaker, and that/those, used for items that are further away from the speaker, relatively speaking.
Many determiners express a notion of quantification. That is they specify how much or how many of the head noun there are. Here's a list of some common quantifiers:
One kind of determiner that deserves separate attention is the numeral. Numerals appear in one of two forms: cardinal (one, two, three, etc.) and ordinal (first, second, third, etc.). When numerals appear in front of a noun in order to quantify it (two birds, four cats, etc.) they are determiners. Numerals can also appear as independent nouns in their own right. We will return to this point when we examine the structure of noun phrases.
 The technical term for this pointing function is deixis.
 We use the term "numeral" in order to distinguish from linguistic number (singular/plural).