What was the role of Canada in the Second World War?

I'll preface this answer by saying that the allies would have won without Canada, The Big Three of Britain,American, and the USSR did the majority of the work, but that doesn't mean Canada, Australia's, New Zealand's, etc. contribution was meaningless. Over 1 million Canadians served in the various branches of service and they all played a role in helping to defeat the Axis. The contribution Canada made is quite impressive when you consider how small the country was population wise (only about 12 million people at the time). They definitely punched above their weight.

The first real achievement of the Canadian forces was that they participated at all. Canada's army, air force, and navy had been decimated by the Great Depression. In 1931 Canada's professional army (there was a non active militia and a permanent, like the US Army and the National Guard) was only about 4000 strong and had a budget of only 13 million. Canada's navy fared better, as it had recently purchased two modern destroyers (Saguenay and the Skeena). Although some had argued that to cut costs the Navy should be disbanded and put all Canadian ships under British authority, but that never came to fruition. The air force was probably in the worst shape, seeing as it had only about 1000 personal in 1939 and all its aircraft were obsolete, the RCAF as one Canadian Historian put it was a "flying joke". So Canada's first real achievement was being able to field an army, navy,and air force that were among the best in the world.

The Navy was pretty much relegated to a defence role. The Navy played a vital role in the battle of the Atlantic, but didn't really participate in any major naval battles. The air force sent squadrons and volunteer pilots to fight in the battle of Britain and participated in the bombing campaigns over Germany.

Canadian troops first set foot on Europe in the aftermath of the German invasion of France. The First Canadian infantry division landed in France on July 12th and July 13th, but the British thought better of the operation and realized France couldn't be saved, so they evacuated before ever meeting the Germans. The Canadians would not set foot on French soil again until the Dieppe Raid in 1942.

Canadian troops would next see action in Hong Kong. The Canadian government had sent a garrison force to help guard Hong Kong, when the Japanese invaded they ran into the mixed force of British,Canadians, Indians, and local troops. There were two mains allied forces guarding Hong Kong "East Brigade" and "West Brigade", the West Brigade was under the command of Canadian Brigadier General J.K. Lawson. Eventually, despite heavy casualties, the Japanese pushed back the defenders, mainly due to the lack of reinforcements and the lack of air cover for the allies. J.K. Lawson had his HQ surrounded and his last words to his superior officers was that his men and him weren't "going to be killed like rats" Lawson charged out to lead the defence of his HQ personally, he was cut down by Japanese soldiers, he was the first senior Canadian officer to die in battle.

Next was the Dieppe Raid. The allies wanted to determine the difficulties involved in seizing a port after an amphibious invasion. The force was about 6000 strong, consisting primarily of Canadian soldiers, with a smaller British contingent. Dieppe was a rather insignificant town, its port was small and generally useless for large scale supply operations. It was only chosen because it was in range of British air cover and wasn't an obvious choice for a raid. British intelligence had underestimated the forces available to the Germans at Dieppe. The port was guarded by the 302nd German Infantry division and it had the 10th Panzer division in reserve at Amiens. British commandos were supposed to neutralize the Germans coastal batteries but were scattered. As a result the Canadians ran headlong into entrenched Germans, who were able to destroy the few tanks the Canadians brought with them (they belonged to the 14th Canadian tank regiment). The Royal Canadian regiment landed East of Dieppe at a small place called "Puys" they were supposed to flank the Germans and neutralize some German defences at Dieppe, they failed to do so and were almost entirely annihilated by the Germans who had been station at Puys. The Canadians still managed to make some progress, and a few units such as the Cameron Highlanders of Canada managed to penetrate into the small Hamlets surrounding Dieppe, but it was all for naught as the Germans kept up their assault and the Canadians were either evacuated, killed, or captured.

The 1st Canadian Infantry division and the 1st Canadian Tank Brigade would be the next units to see action, as they were sent into the invasion of Sicily. The Canadians were given their own beach on Sicily, and despite losing most of their vehicles at sea, made rapid progress and encountered no real opposition. The first major battle happened at a place called "Agira" where the Canadians battled the entrenched Germans. The Canadians fought hard, but the Herman Goring Panzer division managed to hold Agria. Eventually both sides paused, but where as the Canadians were in no danger (as the Americans and British were making short work of the scattered Germans and Italians in the other parts of the island) the Germans had to evacuate Agria because they feared getting cut off. After that the Canadian units were withdrawn to participate in the assault on Italy's mainland.

I'm going to gloss over much of the Italian campaign and focus on Canada's most important battle of the campaign, the battle of Ortona. Ortona was a port and so held some significance to the allies as a supply area, but the Canadians and most Canadian historians today think that the battle was pointless and that the town could have been bypassed. Ortona was held by the First German parachute division and they were entrenched in Ortona. The Canadians had to cross the Moro river to get to Ortona, and this would be their first major challenge. The initial attack across the river failed, but the second attack on December 8th was successful and the 48th Highlanders of Canada, managed to cross the river. Next the crossroads leading to Ortona were taken, this opened up the way for the Loyal Edmonton regiment to attack Ortona. The Germans had fortified the town heavily. Every building was covered in barbed wire and sandbags, machine gun posts and mines were around every corner. The Germans even had booby trapped objects lying on the street (bottles, Bibles, other random objects could all be wired with explosives). The Canadians developed new tactics to clear the houses. They would push their AT guns right up to the windows and fire directly into the houses, others would throw grenades through the winds then rush through the door spraying machine gun fire in every direction. They also developed a technique called "mouse-holing" where by Canadians would blast through the walls of houses instead of walking in the street where they would be exposed. The Germans eventually retreated when Canadian reinforcements cut off the defenders from resupply. 2300 Canadians had been killed or wounded.

Finally we get to Juno Beach and D-Day. The shore bombardment did very little to help the Canadians who ran headlong into German bunkers and strong points. None the less the Canadians pushed past the beach defences and reached the towns and villages beyond Juno Beach. Despite the popular saying that the Canadians were the only nation to reach its D-Day objectives, the truth is that no nation reached its objectives (which were overtly optimistic and didn't account for the large amount of German resistance). The Canadians would go on to play major roles in the battles for Caen, the Scheltd, and the Liberation of the Netherlands.

The Scheltd was another important campaign for Canada, and arguably the last major battle they really fought in WW2 (they would go on to liberate the Netherlands, but the Germans generally just retreated so there wasn't much large scale fighting).The port of Antwerp was necessary for the Allies to continue their advance into Germany. While Antwerp the city had been liberated, the port was usless as long as the Germans held the small islands that flanked the port known as the "Scheltd Estuary". Before that the Canadians focused on taking the ports of Calais and Bulogne, they did so despite heavy casualties. The 2nd Canadian infantry division started the assault on the Scheldt, they had to push the Germans back across a series of canals. The Albert and Turnhout Canals. The Canadians experienced great success and managed to send the Germans back across both canals in very short order. The Germans were pushed into a pocket (named the "Breskens pocket"), when the Canadians tried to crush the pocket they fought against determined Germans who wanted to fight to the last man. While that battle was going on, another Canadian force was advancing on the Eastern part of the Scheldt. The Calgary Highlanders fought against a determined German group of 2000 paratroopers, the battle lasted until October 10th, when the Canadians launched a massive attack and pushed the Germans back. By October 24th both sides of the Scheldt would be pretty much cleared of German resistance and allied shipping could start coming into Antwerp.

The best source on the Canadian effort in World War Two would be "Maple Leaf Against the Axis" by David J. Bercuson.