Determiners are used to make nouns referential. Determiners in English include Articles (a, the), numbers (two books), demonstratives, and possessives (her book). They can appear as words, clitics, or morphemes.
In syntax, determiners are generally considered the heads of their applicable phrases; while it is tempting to think that head of the phrase "the cat" is the noun (cat), it is in fact the article (the). Therefore "the cat" is a DP (with an NP inside it).