I'm kind of new to stitcher, and my understanding has been that panoramas made of multiple still shots were best for situations that had no movement. However, I have seen several panos where there was a lot of movement (like in a park or a fountain surrounded with people). How is this done? When I tried to do this, my clouds and people had a double image effect, my trees looked blurry, etc)

A number of solutions for you:
 
People
1. Wait until they have moved out of the area where the images overlap
2. Use the Stencil tool in Stitcher to remove the ghosting
 
Clouds/Trees
1. Use either Smartblend or Enblend in your render settings

There must be something else to it. I can't remember all the sites that I've seen before, but here are some examples I found today of what I'm talking about.
 
http://www.vrway.com/vrmag_issue_2 [...] 08854.html
http://www.vrway.com/vrmag_issue_2 [...] 09251.html
http://www.vrway.com/vrmag_hotlist [...] 10265.html
http://www.vrway.com/vrmag_hotlist [...] 10448.html
http://www.vrway.com/vrmag_hotlist [...] 10449.html
 
There is some serious movement all over the place in these panos (especially the last one). How do they do that?

It's called skill. It's an acquired talent. Read the forums, search the internet for info on Photoshop and learn the techniques. Have a small project (say removing one person from an overlapped image) and work on it until you get it right. Then make the project bigger and move on.
 
There's no easy "make a perfect image" button in any software.

saltheart
This is where having different lenses for different situations can be invaluable.
 
A 50mm lens needs 15 to 20 shots (I'm guessing) to cover the interior of a sphere. There would be lots of overlapping seams that would need work with RealViz Stitcher's stencil tool and editing with Photoshop
 
Something like a Nikkor 10.5mm or Sigma 10mm would only need about 6 shots around horizontal plus a Zenith and probably a Nadir.  
 
A Sigma 8mm traditionally needs 4 plus a Zenith and maybe a Nadir shot
 
A Coastal Optics 4.88mm on a dSLR or a Nikon Coolpix 8700 with an FC-E9 Fisheye Lens Convertor only needs 2 shots with one seam all the way around the sphere like the longitude line that separates the eastern and western hemispheres of the earth. Its very easy to visualize the seam and wait until nothing is moving across the seam to take the shot.
 
There's generally a sacrifice in sharpness and resolution with the wider lenses and the Coolpix sensors are a fraction of the size of a dSLR sensor, but they can overcome or reduce problems in many situations
 
But, like Andrew said, it still comes down to the skill, talent and experience of the photographer.

I bought a Canon EFS 10-22mm, but I learned too late that there is a 1.6 conversion factor that means it's really a 16-35mm. Ahh...the price of ignorance. Anyway, according to my Stitcher manual, 35mm will require 50 pictures, while 16mm will require from 14-18 pictures.
Regardless, I still don't understand how to accommodate scenes with hundreds of points of movement. Take the concert picture for instance. You can't wait for the people to move out of the overlap area.  
Oh well, I guess it will just take lots of study and practice.

saltheart
It probably won't take 50 shots with the Canon 10-22mm.  
 
At 10mm the lens has a Field of View of 107° diagonally
 
With the camera in portrait position, more like 18 to 22, with 8-10 tilted up 30°, 8-10 tilted down 30° plus a Zenith and maybe a Nadir  
 
I have one too, but only use it for still photos, not virtual images.
 
Depending on the scene, you definitely CAN wait until people move out of the overlap.  
 
I've done it before, shooting a hotel lobby with people coming and going, but it was with my Coolpix 5400/FC-E9 and there was only one seam to watch.
 
I just got a Sigma 8mm f/3.5 which will need 4 around horizontal plus a Zenith.
 
Scenes like the concert are just tons & tons of work. And are possible but very difficult to do virtual photos of.  
 
Hotels & Real Estate are by far the most common use of virtual photography and the shots are almost always taken with no people in them