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Since the surface of a sphere is different from a(n Eucleadian) plane, are there shapes similar/analogous to Eucledian plane shapes? Are there analogues of a square, a circle, a star polygon, etc? If yes, how do they look, what are they called, and what are the properties? Are they considered 2 dimensional?

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There are spherical triangles, spherical polygons... –  J. M. May 2 '11 at 15:51
    
spherical geometry and trigonometry have been used for a long time by astronomers and sailors –  yoyo May 2 '11 at 20:47
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As J.M. mentions there are. The nicest examples are, IMHO, the spherical triangles and I especially like the idea of a triangle with a base segment on the equator and the two other sides meeting at the North Pole is the classic example of their interesting behaviour. The two base angles are both rightangles, so whatever the angle at the North Pole you end up with an angle sum bigger than 180 degrees, and so on. I won't go on as this can be found in many different places and is fun to search out for yourself.

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