Learn Russian: Благодаря новому процессору компьютер работает намного быстрее. - Thanks to the new processor, the computer works much faster.

To say "much" before a comparative word like быстрее (faster), Russian requires намного or гораздо. You cannot use много by itself to intensify a comparison; много just means "a lot" or "many" when talking about quantities.

Could I use гораздо instead of намного in this sentence?

Yes, absolutely! ...компьютер работает гораздо быстрее means exactly the same thing. Both намного and гораздо are perfectly interchangeable here for saying "much faster."

Why are the words новому процессору in the Dative case?

They follow the preposition благодаря (thanks to). This preposition always requires the Dative case. It helps to remember that the related verb благодарить (to thank) literally translates to giving thanks to someone or something, which naturally triggers the Dative.

Is быстрее an adjective or an adverb here?

In this sentence, it's an adverb because it describes the verb работает (how the computer works). However, in Russian, the comparative form быстрее looks exactly the same whether it's used as an adverb ("works faster") or an adjective ("a faster computer").