Skiing in the Anchorage area consists of basically 4 ski areas.
Alyeska, the largest of the 4, is a challenging mid-sized hill that caters to the intermediate to expert skier. Most of the upper mountain is open bowl skiing, which gets interesting in the frequent flat light. You get used to it. There are some nice steeps, and they get an average of about 550 inches of snow at the top each year. Skiing usually runs from late November to late May, with camps happening in June most years. The base elevation is only about 250 feet above sea level, so conditions can vary from light powder to ice. Alyeska tends to make solid skiers, and they generally like to ski fast. There is an active race program on the hill (I'm involved with the Masters program), and there's some freestyle terrain as well (halfpipe and terrain park). You can get discounted lift tickets on base.
On Elmendorf AFB, there is Hillberg Ski Area, a beginner ski area with a chair lift, rope tow, and t-bar. A great (and cheap) place to bring the family, learn to ski, or goof off on the tubing hill. Sometimes they can have the best conditions in the area during the early season. I run the intramural racing program there. They have a small terrain park and we're looking at building other terrain features for next season. The pro shop has good deals on gear, and a season pass is less than $200. Ft Richardson has Dyea ski area, but even tiny Hillberg makes it look small.
The main ski area in Anchorage is Hilltop, which is slightly larger than Hillberg and more expensive. The majority of the hill is pretty flat, and they cater to the freestyle crowd and school age kids. It can get pretty "Lord of the Flies" at times. Honestly, if you're going to spend the money, Alyeska is a better bet.
The Anchorage Ski Club owns Alpenglow, a crunchy ski area just above Anchorage. Run on a shoestring, it has some great runs but can be pretty hit or miss. When it's good, it's great, and when it's not, it's painful. Minimal grooming and no snowmaking. Worth the trip after a big dump. Great people.
Besides the resorts, you can always do backcountry in Hatchers Pass or Turnagain Pass. Heliskiing is available in Girdwood or Valdez, and cat skiing operations are another option.
The low elevation and proximity to the water affects all of the ski areas. The weather varies from -20F to 40F during the peak ski season, rain and snow, lots of overcast days, and not a whole lot of light. You learn to cope and enjoy skiing in just about anything. I wouldn't have it any other way.
If you have any specific questions, let me know.