Would you like to buy this new item? (Identify the word class of the underlined word 'this'.)

In this sentence, 'this' is used before the noun phrase 'new item' to specify which item is being referred to. Words that introduce nouns and specify them (like articles, demonstratives, possessives) are determiners. Hence 'this' is a determiner here. (In some traditional grammars it is called a demonstrative adjective, but modern linguistics classifies it as a determiner.)