1) In order to attenuate peremptory requests, orders and suggestions. It is used to express courtesy and politeness (condizionale di cortesia).
Vorrei che tu venissi con noi questa sera. I would like you to come with us tonight.
Io avrei un consiglio per te. (I would have/I have) a piece of advice for you.
Vorrei una fetta di torta. I would like a piece/slice of cake.
NB:Volere in the conditional, as shown in the example above, is normally used to politely ask for something at a restaurant, bar, or any sort of shop.
2) The conditional is used often in journalism/in the news to highlight the uncertainty of the information that is about to be disclosed. In Italian, this is referred to as condizionale di dissociazione (The Dissociative Conditional) which further emphasizes that the speaker is relaying information that is most likely true, but that he or she is not absolutely certain of.