There were variations over time, but in general there was an uneasy standoff between Mussolini's Italy and the Vatican City. Some major points of inflection are:
Lateran Accords
In the signing of the Lateran Accords (or Lateran Concordat), Mussolini achieved a great victory by ending "the Roman Question" that had lingered from the 1800s regarding the status of Rome both as the capital of the newly unified Italy as well as the seat of Catholicism. This treaty created the Vatican City as its own nation and regulated the interactions between Italy and the new nation. However, both parties were leery of the other and occasionally annoyed at what was seen as interference by the other party.
Invasion of Abyssinia
The Lateran Accords required strict neutrality in foreign affairs but allowed the papacy to speak on issues of morality. The first big test of this duality was the Italian invasion of Abyssinia, now known as Ethiopia. Pius XI (pontiff from 1922 until 1939) clashed with Mussolini's representatives. His top aide in foreign affairs was Cardinal Secretary of State Eugenio Pacelli--later to become Pope Pius XII. Robert Ventresca highlights the differences in temperament between these two figures in his biography of the latter man, and describes their partnership in eloquent terms. To simplify things quite a lot, Pius XI was a man of passion and emotion, while Pacelli was a man of diplomacy and rather reserved. Together, they attempted to keep Italy from invading, but with little effect. (David Kertzer has a different interpretation of the same evidence in The Pope and Mussolini, alleging a conspiracy between the two nations that resulted in Mussolini buying the silence of Pius XI. I find his argument unconvincing, especially in that his biases color his conclusions) Italy invaded and the Vatican's disapproval was largely behind the scenes.
Mit Brennender Sorge
The promulgation of Mit Brennender Sorge (issued by Pius XI, with its authorship largely credited to Pacelli) strained relations further. Its critique of fascism was keenly felt both in Germany and in other areas where fascism was popular--particularly Italy and Spain. In all three countries its distribution was banned and copies were seized. Its critique of theories of racial superiority and the excesses of the State in fascist systems was a real test of how the Vatican could speak on morality while maintaining the treaty requirements of neutrality in foreign affairs.
Racial laws in Italy
The passing of racial laws in Italy in 1938 strained the relationship between the Vatican and Mussolini's Italy even further. The Pope's Jews by Gordon Thomas covers this fairly well (though I have some problems with its style). There was never quite a full rupture between the two governments, and Ventresca credits this to Pacelli's style of engagement rather than confrontation with an eye to a pragmatic approach to mitigate the problems people were facing.
Election of Pius XII
Pacelli's election wasn't a shock, but how he would respond to the rapidly unfolding events of 1939 was a bit of a mystery. For a time, Mussolini's government thought they could engage with Pius XII on a more favorable basis than they had with the increasingly combative Pius XI. The pontiff's first encyclical--*Summi Pontificatus was issued in October of 1939), and its approach combined a number of Pius XII's convictions, including
Throughout the rest of WWII, Mussolini's government had tense relations with the Vatican City until Mussolini's downfall and eventual death.
I hope that answers your question. Followup questions from OP and others are always encouraged. I can go into a fair bit of depth on most of these topics, but I didn't know how much detail you wanted on any one of them.