Determiners typically indicate whether the referent of the noun phrase is assumed to be known to the hearer/reader. For example, the noun phrase the car in an utterance like (1) would normally only be used if the hearer can be assumed to identify what specific car the speaker has in mind.
(1) The car was sold to a little old lady.
The second noun phrase (a little old lady) can be used without any assumption that the hearer should be able to identify a specific lady as the buyer of the car. The crucial difference between the two noun phrases is that the first one has a definite article functioning as the determiner, whereas the second one has an indefinite article functioning as a determiner.
Different types of determiners
The articles (the definite article the, and the indefinite article a/an) are prototypical determiners. However, many other words can function as determiners in a noun phrase. Below is a list of some of the more common types, with examples.
some, any, no, many, much, few, little, a great deal of, a great number of
Possessive pronouns and
my, your, his, her, its,
this, these, that, those
Determiners are sensitive to the number of the noun that they combine with. Demonstrative pronouns, for example, vary in form depending on whether they combine with a singular or a plural noun.
(2) this book (singular) — these books (plural)
(3) that book (singular) — those books (plural)
Countable and uncountable nouns differ with respect to what determiners they can combine with. See the following links for a discussion of differences between countable and uncountable nouns and for other difficulties related to the use of quantifiers in particular.