It seems that they started out as fairly inexperienced divisions. However gained experience quickly early in the war. The real question is that on top of replacing losses to existing divisions, they raised A LOT of divisions post '42 which would seem to defy the elite status of the best men with the best training/equipment.
You can't compare the SS from the early days to the Waffen-SS of 1945. The organization started as a personal guard for Hitler and a crowd control force for political rallies in the 30's. They were lightly armed and highly mobile, to be able to react quickly to civilian, internal threats. There is a huge difference between the SS Wiking fighting on the southern front in Ukraine in 1941-1942 and the Waffen SS men that ended up in the trenches of Kurland. The latter were more auxiliary infantry than the elite and highly mobile units from the Blitzkrieg of 1941. The units that fought in Berlin in 1945 were desperate and battle hardened and fought a defensive battle. That gave them better tactical opportunities.
When Heinz Guderian and Erich von Manstein designed the Blitzkrieg operational concept in the 30's on the Munster training grounds, the highly mobile and ideologically indoctrinated SS crowd control forces, proved to be very useful. They fitted the requirements: they already operated in columns and were the first totally mobilized infantry units. Given some new weaponry and making smart use of close air support, they were organized to concentrate firepower on short stretches of the front line, force a breakthrough and penetrate deep into enemy territory and cutting off enemy supply and communication lines. Thus creating chaos in the enemy's ranks and deny the enemy resources and the will to maintain fighting.
This operational concept was fit for the Western European space, ideal for a smaller force in a limited space. Because of the success of the Westfeldzug, the operational concept became famous as were the units that fought the campaign. And those were for a decent part SS units. In the Blitzkrieg Legend, KH Frieser describes that the successes of 1940 were for a large part the result of luck, coincidence and a revolt among the German General staff. The conservative Wehrmacht Generals did not want this concept but a few SS Generals did it anyway. Hitler liked the success and demanded to implement the operational concept on the much larger Russian space. The SS made a name for itself, by defying the conservative Generals and winning in the west. But for this to work in the east, much more equipment and manpower was needed.
Being a volunteer force and very motivated, they were a tough bunch to beat. The Waffen-SS in the early war years was supplied with the best equipment. In the east Waffen-SS units were used as a reserve to be thrown into battle at the decisive moment, when regular Wehrmacht units already had softened the target. Their battlefield performance was excellent and most German Generals were master tacticians. This contributed to their legendary status: always the underdog and always fighting in the battlefield hot spots. Their maneuvering skills were brilliant, but due to a lack of means, could not beat the Russians on the strategic level.
So when the Allies were confronted with the SS units after the Normandy invasion, they encountered a very experienced force, used to fight against superior forces in brutal circumstances in the east. They relied on superior maneuvering instead of superior logistics, often outnumbered and very experienced.
In the first 2 to 3 years of WW2, very specific recruiting standards were maintained: blond, tall, strong, blue eyes, no Jewish/Slavic ancestors. If a SS Mann wanted to get married, they investigated the bloodlines of his future spouse. A few years back I saw the documents of the investigation of Joseph Mengele's wife in the Bundesarchiv in the former Prussian Kriegschule and HQ of the SS Leibstandarte AH in Berlin. Before he was a doctor in Auschwitz, he was attached to a battalion of SS Wiking, that fought on the southern front and the Caucasus. Those "pure" racial standards gave them an elite status as well.
But in 44 and 45 the Waffen-SS ended up being the German Foreign Legion, taking every able bodied man in their ranks, even Slavic men who only a few years earlier were seen as inferior. The Germans had a serious lack of manpower after the losses on the eastern front and were not in a position to be selective.
So they required the mythical status in the Blitzkrieg campaign in 1940 and were given a special role during the campaigns in the east from June 1941 on. This special role and the issuing of the newest and best equipment gave them a fierce reputation. Later on, e.g. during the Battle of Berlin, they weren't that well equipped anymore, but years of battlefield experience, a favorable defensive fight and nowhere to flee to, made them a bunch of fierce fighters and earned them the elite status.
Because a lot of the Waffen SS Divisions were "elite". As you can imagine a big factor for the combat effectivness of a soldier is his determination, conviction and overall moral. The people who fought in the first Waffen SS divisions were mostly volounteers. This is important, these people choose to go to combat and were willing. The criteria for SS troops was tough especially for the Leibstandarte ( LSSAH ) they were above average in physical condition of the Wehrmacht.
As pointed out before the overall moral of a soldier is one of the single most important factors. Most units get destroyed ( for example captured ) when their lines break or they get outflanked, this happens primarly if big amount of soldiers surrender at the same time, like for example an entire company. These occurence were far less likely in these Divisions.
When people think about "elite" units in World War 2 terms they think it is mostly about the soldiers beeing good marksmen or having good equipment or training. This is a flawed assessment. More important than beeing good with the rifle is that soldiers hold their positions and don't surrender. This enables commanders to plan better because they have less crisis situations, stuborn soldiers naturally inflict more casualties. The overall tactics were the same for most Wehrmacht units, artillery et cetera had the same operating procedures. Training was also mostly the same. What worked was obviously used for all units...
A good example to illustrate my point is the 12 SS division Hitlerjugend. This division is unique since its infantry manpower were mainly 17 and 18 year old. Obviously these soldiers could not have more training or more experience than other formations since the unit was only some months old and none of the hitler youth men had combat experience. The point where these soldiers excelled was moral and fanatism.
During Normandy this division outperformed a lot of other divisions and hold their ground against a lot of manpower and allied ressources. They never surrendered in great numbers and inflicted serious casualties despite beeing in an unfavorable position. Since this division was green ( except many officers) the reason for the good overall performance can hardly be experience and training. In the american sector a lot of infantry divisions surrendered in big numbers, many of these divisons used Ostlegionen. Ostlegion were soldiers drafted from countries like Russia and Ukraine, obviously these soldiers were not as willing to die for Hitler, so many of the surrendered and compromised the defensiv positions of their adjacent units. The rest is basic math, a soldier who surrenders inflicts less casualties as a soldier who shoots until he dies. Another example would be the 16th Luftwaffefeld, the unit was heavily hit by bombardment during Goodwood and basically got destroyed when big chunks of soldiers jsut surrendered, their will was just broken. The Hitlerjugend which lay next to the 16th was also quite frequently hit by bomberstreams but didn't lose its cohesion.
The summary of my point boils down to: The moral of a soldier is of utmost importance and here is were Waffen SS divisions excelled. These soldiers were fanatics and brainwashed. As sad as its sounds if you want the best soldiers you want people who got brainwashed since young age. Obviously it helped that the tactics, training and equipment of the Wehrmacht was good.
Regarding literature, Reynolds book about the SS units in Normandy was pretty good but focused more on day to day action. If you want to read more about the 12 SS in particular Hubert Meyers book is the most extensive on the subject. He was part of the unit, keep that in mind if you read the book. A lot of other historians talk about the 12th in their books about Normandy, Hastings for example.
As someone with a fairly good broad knowledge of WW2 this has been very interesting and informative to get a more in depth insight. I have a bit of a follow up for you all.
From what has been posted, it seems the most significant factor of SS troops being "elite" was their morale and aggressiveness. Was this markedly higher than regular Heer soldiers. Also, did the slow drain on manpower affect the SS or Heer more over the course of the war?
I think it is worth pointing out here that not all Waffen SS units were military elites.
The 36th (Dirlewanger) Division was simply a motley collection of criminals. Dirlewanger himself was a criminal and a peadophile, and his "Division" spent most of the war on "anti partisan" operations, which was a euphemism for conducting a campaign of murder, rape and pillage against the civilian population of occupied eastern Europe. On the rare occasion that the Dirlewanger division was engaged in actual combat, they suffered severely.
The Kaminski Brigade was similarly an "anti-partisan" brigade - in other words they conducted campaigns of murder and terror against civilian populations while being little short of an ill-disciplined gang of brigands of virtually zero actual military value.
The Free India Legion was initially conceived as being composed of Indian POW's who would switch allegiance and lead an invasion of British India through modern day Iran. At its height, the India Legion numbered about 2,000 recruits making the ambition to invade India risible. The Legion barely saw combat and remained a drain on precious resources and equipment while being little more than another bizarre Nazi fantasy formation of dubious loyalty.
The British Free Corps was a bit of a joke. Never numbering more than enough men to complete a platoon, let alone a corps, they were dismissed from the front line before they saw action by sympathetic German officers.
All in all, the Waffen SS were a pretty mixed bag, and quite well reflected the dichotomy of the Prussian military tradition on one hand, and chaotic, perverted, and nihilistic Nazi ideology on the other.
Edit: The SS: A New History by Adrian Weale is a good primer on the SS.