So the Germans strategy in Italy was to basically hold the allies at bay and make them lose as many people as possible. The Germans knew they didn't posses the power to actually kick the allies out of Italy, but they were determined to make the allies lose as many people as possible. And Italy is perfect ground for this, being that it is very mountainous with plenty of defensible locations. the German commander in the area named Albert Kesselring was a rather talented commander and built a number of defensive lines designed to stall the allies. One of these lines was called the "Gustav line" and was just south of Rome. There was another one called the gothic line in the north of Italy. If you wanted to get to Rome, you needed to take certain roads, however the Germans and the harsh weather made it so that two of these three roads were unusable, so the only one left was called Highway 6, which led into the Liri valley. If you wanted to get through the valley, Monte Cassino would need to be taken, otherwise the Germans could simply fire on whoever was in the valley.

Anyways, Monte Cassino was a hill top fortress that contained an old monastery and the Germans had fortified it heavily. Monte Cassino anchored the western end of the winter line and was thus heavily defended. The allies tried to smash through the valley rather than going up the actual Cassino heights, this failed and cost an American division around 1000 casualties. This meant that the American commander Mark Clark would have to take the heavily fortified hill.

The allies tried three times. The first battle saw the Americans get repulsed easily. The second battle saw the New Zealand and British Indian troops try and take the hill, this time the allies were better prepared as bombed the hill relentlessly, turning the monastery into rubble. But again the German defenders beat back the allied advance. The third battle saw the New Zealanders and Indians try another assault, again they failed despite even more air support than last time.

Finally the allies broke through, and it was truly a multi national effort. The polish second corps would move around Cassino and assault from the North, while the French moved infiltrated the mountains from the south, the Canadian armour could than move through the Liri valley, while all of this was going on the Americans would be attacking on the far west coast in an attempt to link up with the Anzio beachhead. The plan worked, Cassino fell and the Germans retreated. Rome was captured.

There is some controversy though, see there was also the Anzio bridgehead. The allies had tried landing at Anzio behind the German lines but the Germans had kept the allies bottled up on the beaches. The plan was to break through the Gustav line on both sides while breaking out of the Anzio bridgehead and this encircling the Germans. However some claim that Clark captured rome instead of finishing the encircling of the German army, but that's heavily debated and meant for another post.

A nice digestible read on this would be either John Keegan's "The Second World War" or Max Hasting's "All he'll let loose: The world at war 1939-1945"