Having read--and even written--that Wolfram Alpha is some sort of cyber wonderbrain, I must say that now that I am using it I feel a bit underwhelmed. I do not know what I was expecting, but an online Farmer's (and Geek's) Almanac isn't it.
Looking at the examples on the site, what WA seems to do best is pull up facts and try to piece them together. If you need to know about gravitational acceleration, which Wolfram Alpha tells me is "total field | 9.78936490256 m/s^2 (meters per second squared)" at my house, this is a great tool.
Or maybe you're interested in "n-grams 'it was the best of times it was the worst of times'" I am not sure why you would be, but someone must be as this is a real example that Wolfram offers to explain what Wolfram Alpha can do.
It also offers access to much less geeky information, such as exchange rates, Internet lookups, weather, and about a zillion other things. Venture very far off the examples, however, and you'll find that no map of California cities or counties is available and that entering "SQL" will tell you everything you want to know about the San Carlos, CA airport, but nothing about structured query language.
If you spend enough time looking at the examples you will learn that Wolfram Alpha is much better with numbers than worlds, which isn't surprising since it runs atop Wolfram's Mathematica equation processing engine.
There is a lot that Wolfram Alpha can tell you, though sometimes it leaves things about. After telling me about the orbit of Halley's Comet and when it was discovered and how long it takes the circle the Sun, Wolfram Alpha was silent on the obvious question: When will the comet return?
As a work in progress, Wolfram Alpha is likely to have all manner of lapses such as that, though they don't take away from what the site does offer. Weather, sports, people, materials, math, socioeconomic data, engineering, transportation, physics, chemistry, and a variety of other topics are all there.
Much of this information is of particular interest to business or is about business. You can find out about companies, but don't expect to dig too deeply.
Wolfram Alpha seems to be more of an example of what Mathematica can do than a serious attempt at a competitive search tool. It really is more of a place to find very specific facts, figures, and do calculations. Entering my birthday, for example, revealed that I am 49.96 years old.
The company calls Wolfram Alpha a "computational knowledge engine," which is probably the best description, but only after you've actually seen Wolfram Alpha in action so you know a "computational knowledge engine" when you see one.
I believe Wolfram Alpha will be popular with people who find certain features that the simple search query-style user interface is well suited for. Or who understand what you get when you enter a city name or a date or whatever else they are interested in.
The best approach to Wolfram Alpha, for a newcomer, is to explore, find things that interest you, and try to remember them so when you need that sort of information in the future you'll know Wolfram Alpha has the answer.