*note, this answer covers the German Waffen-SS units, not the foreign Waffen-SS units, who were more variable in quality, but whose formations had less to do with the front and more to do with rear-line security and increasing the power of the SS in occupied Europe.

Although the Waffen-SS gained a reputation during the war for ferociousness and tenacity, and this reputation grew significantly during the postwar period, their actual combat success was more mixed and prosaic. Some Waffen-SS formations fought rather well, but others' performance was not terribly impressive. The issue of actual combat effectiveness of the Waffen-SS is rather difficult to parse out given the many different variables involved. For example, while Manstein's memoirs praised the performance of Totenkopf on the Eastern Front, but reports from both Manstein and his fellow officers in the 1940 French campaign tended to condemn this division's performance. Fedor von Bock noted in his war diary in the Spring of 1940 after visiting a SS motorized regiment under his command that:

The combat training of the non-commissioned officers and men of the SS is inadequate; it will cost a lot of blood! Shame about the splendid manpower material.

Wartime experience bore out Bock's prediction. Waffen-SS troops typically suffered proportionally higher casualties than their Heer counterparts, and one of the Leitmotifs of postwar memoirs by Heer generals was that the wastage of these fine recruits was the result of dilettantish approach to military science on the part of their officers.

The training regimen of the Waffen-SS tended to be roughly similiar to the Heer, but with several notable differences. The Waffen-SS's recruits were volunteers, in contrast to the conscript Heer, and in the early war period, SS recruits had more stringent physical requirements for service. Wartime volunteers for the Waffen-SS tended to grow during the war, only falling off during 1944. This was partly a function of the outsized role the Waffen-SS played in wartime propaganda as the "fire-brigade" of the front, but also reflected the growing role of the SS inside the Third Reich's state apparatus. For young careerist Germans, joining the SS also meant that they would have a privileged access to the civil service postwar if Germany had prevailed. Thus the Waffen-SS recruit base, especially in the first half of the war, was more homogeneous than their Heer counterparts and the SS often flaunted its agreement with the Heer to keep to its established quotas that limited its recruiting among the German population. This further exacerbated relations between the Heer and the SS, especially since the Waffen-SS benefited from the SS's ascendancy within the Third Reich's governmental structure. As such, the Waffen-SS formations tended to get far more military equipment than their Heer analogues. Waffen-SS divisions had a higher establishment figure than Heer divisions (ca. 20000 vs 16500) and a much greater degree of motorization.

The Waffen-SS's self-styled image of an elite of the New Order was a significant aspect of its training regimen. Far more than the Heer, Waffen-SS training emphasized ideological zeal and commitment to the ideals of National Socialism. For younger German volunteers, this was not too far out of line with their earlier socialization within the Third Reich. Institutions like the Hitler Jugend and the Reichsarbeitsdienst also emphasized physical labor and ideological training, and was thus congruent with their experience within the SS training. The Waffen-SS also discouraged the barriers between officers, NCO, and enlisted men that was typical in the Prussian-dominated structures of the Heer. The result was the creation of a military culture within the Waffen-SS that promoted a more cohesive group identity and a greater ideological commitment than other conventional German military formations.

This cohesiveness did come at a cost though for the Waffen-SS. A number of Heer officers tended to disparage the professional capabilities of Waffen-SS officers and critiqued their lack of knowledge of the finer points of military science. Field Marshal von Rundstedt would once castigate SS-Oberst-Gruppenführer Sepp Dietrich's inability to understand the military situation even with the help of a map. In a February 1943 inspection of SS-Leibstandarte division, Major Ulrich de Maizière echoed von Rundstedt's disdain for the Waffen-SS's military dilettantism:

The commanders of this Waffen-SS division did not seem to realize that brave and ideologically misguided young men were being senselessly sacrificed through insane arrogance and a lofty disdain for sound training. Belief in the Führer was more important to them than professional ability. Shocked and sobered by the experience, I returned to headquarters where I was given an opportunity to report my impressions to the chief of the general staff.

Moreover, overemphasis upon National Socialist ideological training also reinforced the brutality of the Waffen-SS towards both enemy prisoners and conquered populations. While the Heer was not above such crimes, as a part of the SS, Waffen-SS units had a parallel responsibility for security and ethnic cleansing operations which were part of the SS's growing responsibilities.

The prosaic reality of the Waffen-SS was that while its relatively large formations often had decent equipment and above-average recruits, the institution as a whole was not that impressive from the stand point of military effectiveness. In some cases during the war such as the Normandy campaign, ideological fervor and commitment could sustain these Waffen-SS formations in the face of Allied numerical and technical superiority. But this ideological indoctrination often came at the price of sound military training.

Sources

Bartov, Omer. Hitler's Army: Soldiers, Nazis, and War in the Third Reich. New York: Oxford University Press, 1991.

Militärgeschichtliches Forschungsamt. Germany and the Second World War. / Vol. 5, Organization and mobilization of the German sphere of power. Part 1, Wartime administration, economy, and manpower resources 1939-1941. Oxford: Clarendon Press, 2000.

_. Germany and the Second World War. / Vol. 5, Organization and mobilization of the German sphere of power. Part 2, Wartime administration, economy, and manpower resources 1942-1944/5. Oxford: Clarendon Press, 2003.

_. Germany and the Second World War. : Vol. 9/1, German wartime society 1939-1945 politicization, disintegration, and the struggle for survival Oxford: Clarendon Press, 2008.

Adding to the other answer : The French Legion and the Navy Seals are not comparable in being a 'elite force'. The Navy Seals (or DEVGRU, the former Seal Team Six) are a unit consisting of elite soldiers that are recruited from within the armed forces of the United States : First you become a soldier, then a Seal. They are highly specialised in their training and focussing on (covert) operations like sabotage, hostage/VIP rescue, reconnaissance and direct hits, with a focus on deployments in 'wet' areas (The United States Special Forces are notoriously messily organised and it's hard to tell who does exactly what, but thats the rough idea)

The French Foreign Legion is a part of the French Armed forces with the core of the enlisted men being foreign. They are on the verge of mercenaries (paid men from other nations fighting for France) and regular soldiers (which they are seen as in terms on international law). They are not necessarily an elite force and do not have the same kind of special training the Seals have. They do however have quite a reputation as a unit with an out of the ordinary stenght in combat. Most of the recruits are trained (former) soldiers from other nations and the fact that they left their 'old life' in order to fight for another country shows they are a special kind of force. They have a high corps spirit compared to other units.

An 'equivalent' so to say in the US armed forces could be the Marines. They aren't a 'true' special force either but have a made a name as an 'elite' force for themselves to and are regarded as a unit/force you can deploy when the shit is about to hit the fan. When the US President says 'Send the marines' the French would say 'Send the Legion'.