Although at first sight one would believe that "Stacy" is a noun, being the name of a person, the fact that its form is possessive, as indicated by "'s", makes it an adjective that modifies the noun "dog" (i.e. it tells us whose dog it is). "Beautiful", of course, is another adjective, also modifying the noun "dog". "Needs" is a verb, and "clothes" is a noun. "Stacy's beautiful dog" is the subject of the sentence; "needs" is the verb, and "clothes" is the direct object of this verb.