Wondering what a certain technique is called

Long exposure.

 

In camera terms you might also call it bulb exposure. The premise is that you set your camera to shutter priority mode and then set the shutter to a number at least a second or longer, for light artwork it can be up to 30 seconds. If you want to control just exactly how long the shutter stays open, then look into bulb exposure mode.

As for your second question, it doesn't work that well in the daytime, ends up looking like a blurry blob of nothing.

 

You can easily perform the same technique in bright daylight if you use a ND (neutral density) filter on your lens and have the camera mounted on a steady tripod. Depending on how long of an exposure you're looking for, and at what aperture, you may need a very dark filter, perhaps on the order of 8 or 10 stops. This technique is also very useful for removing people from an image, while the background remains clear. An example might be wanting a picture of the Lincoln Memorial without the tourists in the shot.

 

Yes you will need a tripod for the waterfall shots since the best blur comes at around shutter speeds of ~1/2sec or slower. Image stabilization on your lens will help, but at 1/2 second it's still going to be blurry due to camera motion. Take many repetitions of the same shot, so you can pick the sharpest one afterwards.

Some alternatives are to try finding a rock or something to rest the camera on, as it will be lying still during the expsoure.

Ruahrc

 

Note however that this is isn't suited for the "multiple exposures of one person with flash" technique, since you'd need pretty strong flash to work through that heavy ND.

 

also:

if you have Photoshop, you can also take around 20-30 shots on a tripod of the scene with people in it (same exposure and everything)

in Bridge, select the file, then go to Tools>Photoshop>Load Files into Photoshop Layers
select all of the layers, then go to Layer>Smart Object>Convert into Smart Object
finally, go to Layer>Smart Object>Stack Mode>Median

usually removes 90% or so

(the people that don't move much throughout the shots will not really be removed)

 

"Painting with Light" or "Light Painting"

http://en.wikipedia.org/wiki/Light_painting

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