Pros: Excellent mountain restaurants, good intermediate terrain, good lift system, pleasant town.
Cons: Limited, no true beginner terrain, little very challenging terrain, expensive drinks.
I haven't visited as a tourist yet (so to speak), however, have spent a few weeks doing temporary instructor jobs and have had a couple of good free-skis to check out the slopes and a wander around the town.
The skiing at Courmayeur is nothing special. This is a small-to-mid sized resort (by European standards), and there is little variation. There are plenty of wide open, steep red runs which are well groomed, and a few nice blue runs around the resort. The only truly challenging run is a 1.5km bumps run (with no escape route) - make sure your legs are well prepared before dropping into that one! I couldn't even find a terrain park, but there are a few features dotted around. There are, however, a few options for those looking to head out for a spot of alpine touring on the large glaciers lining the many 4,000m plus peaks surrounding the resort.
Courmayeur is not designed for beginners. A new skier, if sporty and with little fear, does have plenty of easy intermediate terrain to move onto quite quickly, but for nervous or uncoordinated beginners, the learning curve is steep (as are the beginner runs).
The lift system is generally good. A gondola and cable car service the slopes from the town, providing two main upload points which meet at the same mid-station. The chairs are mostly high speed quads, with a new six pack serving some of the best terrain. There is a further gondola and cable car higher up and two old double chairs from around the mid-station (the only non-detachables in the resort).
Courmayeur has some excellent restaurants on the mountain, providing really good food at reasonable prices. When in Italy you naturally have to have the pasta (great carbo boost for skiing too) and you won't be disappointed with any of the offerings.
The town is a collection of pretty stone buildings along a main pedestrianized zone, with further buildings spreading from the centre. More excellent restaurants are to be found here, as well as a few nice bars. Bar Roma is one of my favourites, despite the €8 price tag for a large beer, as there is always a good spread of excellent food at no extra cost (it explains the high price tag though - something that is ever present in European resorts anyway). Away from food and drink, there is also a small selection of good ski shops, and a large selection of fashion stores (again, we are in Italy...)
All in all, a very pleasant resort for a relaxing week of skiing, it's just not particularly big and doesn't offer great variation. The views are phenomenal though!