Hi Spiderjon. I'm an English skier responding to the US responses:
Yes, many Alpine ski areas are enormous, sometimes covering 3 or 4 big valleys and several villages, especially in France where the majority of the resorts have been purpose-built since the war in virgin mountain areas (rather than, as in Switzerland and Austria, based on existing villages). This also means they are less picturesque, but often have a better balance of beds to slopes so can often be less crowded.
The mountains are generally higher (though further east, in Austria especially, this may not be true). I know one black run (double black diamond is the rough equivalent, though this may vary from area to area: a red run in a tough resort might be a black in an easier one) which is about 7,000ft (2,100m) vertical.
Some resorts, such as St. Anton, have been forced by the recent culture of litigious skiing (unfortunately spreading here from the US) to reclassify many of their black runs as off-piste itineraries, which are not groomed or patrolled but are cleared for avalanche danger. This was because they were so often having to be closed because of a few exposed stones or some ice that all the good skiers who wanted to ski them were getting frustrated.
The off-piste is the most different aspect of European skiing. There is no Out of Bounds: you can ski anywhere (except a few National Park areas where you need special permission). However nothing off-piste has been cleared of avalanche risk. You ski it at your own risk, making your own assessment of danger. If you want to ski a 60º slope above a cliff you can. Also consequently there is no sign-posting of off-piste runs and gullies. You just have to know where they are.
Don't expect West Coast US type powder, except if you are very lucky. You'll often need to be able to cope with crud: heavy wind or sun affected "powder" or wet snow or crust. Though a good guide will probably find you decent powder long after most people in the resort think it's all gone.
The consequence of all this is that to get the best out of the off-piste you really need to go with a guide. In many resorts there is a huge amount of very challenging off-piste skiing if you know where it is and which are the best resorts to find it.
Sometimes, as in Chamonix (an old town and several ski areas along the valley - the most famous being Argentiere - linked by a free bus service), there are so many good skiers in these places looking for powder that the obvious stuff gets skied-out within half a day of a snow fall.
Others take a bit longer (for example La Grave or Alagna which are essentially tough off-piste only, though starting this year the latter has one groomed run which outrages the purists). But perhaps the surest is often to go to a resort considered easy that has some good off-piste: but only if you have a guide who knows where it is (But those who know such places aren't keen to divulge where they are) or, as in the Dolomites, where hardly anyone skis the steep-off piste between the pinnacles (But there the snow is chancy and you may find there isn't enough snow to ski these rocky routes). In these places you can often still make the first tracks on a slope a week or two after the last snow. But, as always, especially in these days of climate change and rapidly retreating Alpine glaciers, it helps if you are prepared to walk a little to find it.
Some resorts, such as Tignes / Val d'Isere have off piste guiding companies which will put you into a group of similar standard and this can be very good and relatively cheap (for example TopSki in Val d'Isere, one of the very best companies, charge 65 Euros per day for a full day or you can ski mornings - 9am to 1pm for 50 Euros. Another, smaller but very good company there is Alpine Experience who are all English speakers - some British or Australian - though if you want semi-extreme skiing go to TopSki).
Many (perhaps most) other resorts only have guides who can be privately hired and you have to make up your own group or it will be extremely expensive. In general groups are limited to 6 and if you can make your own group of 6 this will be the cheapest option of all (probably about 50-60 Euros per day each, in my experience - and that's with top guides).