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http://www.bhphotovideo.com/c/product/301942-REG/Matthews_755001_Hollywood_Turtle_Base.html
C stands have nonremovable bases. C+ stands can be removed from the base. The "turtle base" is the typical three-leg bottom of a c-stand. With a C+ stand you can remove the long column and use the turtle base as a very low-to-the-ground junior-pin light stand, or get a junior-to-baby adapter and use it as a small baby light stand.
In the old old old days, a turtle base was a weighted mic stand base that looked like the rounded back of a turtle...kinda like cutting off the top third of a ball. That is still my favorite type of mic stand because it is less likely to trip you.
That has somehow come to mean the curved nature of the base...even if legs. The "C" part is often attributed to "Century Stand".