As a supplement to Santiago's answer, the effectiveness of the Panzer divisions early in the war was mostly based on their mobility and speed of action. It worked like this:
Their armoured regiments would break through the defensive line, if they were facing one at all -- making attacks where they weren't expected was a major part of the trick. Then, since the whole division was motorised, it could move much faster than its opposition, many of whom expected their infantry to march. This let them disrupt the opposition rear areas and hence the supplies and command-and-control of the other side.
Another part of it was that the supply units for the panzer divisions were dedicated to that job and fully motorised, so they could keep up, and supplies and fuel for several days of combat could travel with the division. The idea was that they'd have won by then.
One of the things that made it possible, if risky, to do this with relatively small numbers was that the Panzer division commanders didn't worry about maintaining a continuous front line. They were willing to be isolated inside enemy territory, provided they could remain mobile, because they reckoned that they could move faster than troops could be brought into position to fight them.
The final thing that made this trick work was the German command and control system. The commanders would travel with the Panzers, rather than staying in a distant (and safe) headquarters. This saved a great deal of time needed for communications -- radio usually had to be encrypted Morse Code at the time, handled by specialists, rather than direct speech between commanders and operations officers -- and enabled the Germans to do things much faster than their opponents' command structures could react to them.
Of course, they could not carry an elaborate headquarters with them. The way they coped with that had been an aspect of German tactics since the 19th century. Mission-type tactics consist of telling your subordinate units what you want done, but not giving them orders to do it in a specific way. This simplifies the higher headquarters, and saves the tine needed to prepare more detailed orders. It is very demanding for the junior officers, but the pre-WWII German Army had worked hard on training them, and most of those officers were still in place up to fall 1941.