The most usual answer to this is that the Waffen SS was not very militarily effective at the beginning of the war. However, it became an elite unit by the end of the war.
At the beginning of the war, the Waffen SS did not have good training or good NCOs. The organization had been created more as a police force and its membership was based more on racial and political purity than on military prowess. Because of this, the emphasis in its training had been more political than military.
However, as the war went on, the Waffen SS came to be much better trained. This training, along with its higher levels of morale, helped make it an elite force. In many areas of the Eastern Front, for example, it came to be used as an emergency reserve that would be sent wherever there was serious trouble.
So, the Waffen SS started out ineffective but later became the fighting force that was given the hardest assignments because it was so effective.
I concur with the post above. In addition, the Waffen-SS received the best possible supplies and were more well armed than almost any other units. This added to their morale and their effectiveness. By the time of the invasion of the USSR in 1941, they had become experienced soldiers that were Hitler's pride and joy.
Later in the war, because of heavy losses on the Eastern Front, the Waffen-SS became less elite as they recruited from captive populations, and were much less exclusive as the war went on (especially in terms of their racial policies). In fact, the troops that were the last line of defense protecting Hitler's bunker in Berlin were in fact French SS of the Charlemagne division.
Overall, their fanaticism, strength of arms and gathered experience over the course of the war made them formidable foes. See the defense of Budapest in 1944 as a good example of this.