Well this is a very broad question, naturally, and involves a lot of hypotheticals (and therefore a lot of debate), but lets seize on the 'big' arguments that have been made.
Naturally, first and foremost, Hitler's invasion of Russia is usually deemed a foolish move, however, this is speaking with hindsight. It is more intelligent, I feel, to explore mistakes made after the launch of Barbarossa.
The encirclement and destruction of Russian forces at Kiev is largely considered the greatest single tactical victory of WWII and the worst Strategic blunder. It was wasted effort and bought the Stavka (Soviet high command) time to form a cohesive defense around Moscow and form strategic reserves to ensure the defensive lines integrity. The time, manpower and supplies wasted in the effort to destroy the forces in Kiev - at Hitler's behest - could've been used much more efficiently to go for the throat of the Soviet Regime's military and political power, at a time when the RKKA was still very much off balance.
Changing Stalingrad's significance from a blocking position to a full-blown occupation mission. Case Blue's initial goal was only to seize the oil fields of the Caucasus region; the initial drive to do so was well on its way to accomplishing that. Stalingrad, a key crossing point for potential counter-offensives, was only to be invested by fire and a cordon to cover the drive further south...but something changed in Hitler's psyche - and this in and of itself is a matter of debate. Books on this are numerous and beyond count, but if you want a fairly recent publication that is highly accessible, then look for Antony Beevor's 'Stalingrad.'
Denying Manstein more time to continue his defensive operations and ordering a counter-offensive at Kursk.
At the time leading up to the offensive at Kursk, the German military was already and categorically incapable of attacking on a broad front. This lead Hitler to demand a concentrated thrust; the build up to Kursk required stripping other areas of the broad Russian front of their precious armored and mobile formations to accomplish this. Kursk proved to be a disaster despite the heavy losses inflicted on the Russians, and robbed Manstein (Southern Army Group) and his colleagues in the Center and Northern Army Groups from continuing their successful strategy of 'elastic' defending. That is, giving up tracts of land when necessary to buy time for a mobile counterattack to be delivered, destroying large numbers of the enemy with a well coordinated, operational counteroffensive. The argument made here is that Hitler simply attacked too early, and should've let the year of 1943 go by in a purely defensive state while the strained German war machine licked its wounds.
Arguments for what the Allies could've done better are even more numerous, so I'll leave off with one of the more contentious debates that I am familiar with:
Following the breakthrough from the Normandy sector and the subsequent broad pursuit across France against shattered 7th, 15th and 5th Panzer Army remnants, Eisenhower was forced to call for restraint and limit advances due to crushing fuel shortages and the ever-lengthening supply-line. Fuel was made on a priority basis and as we know he gave priority to the Northern shoulder of the advance to conduct operations in the Scheldt (in an attempt to capture Antwerp intact) and, more infamously, in occupied Holland. The argument generally splits into two schools of thought, first; that the operations in Holland were overambitious, and should've been more realistically planned, and second; that these operations should never have been launched, instead, supply priority should've been given to the Southern shoulder to help expedite an invasion of the German homeland.