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The two fixed points are the foci but these do not define the shape of the ellipse. You also need to know the eccentricity.
There are two points: the foci.
No there can never be a single point. But yes there are two such points called foci( each called focus) that helps to define an ellipse. An ellipse can then be defined as a curve which is actually the locus of all points in a plane,the sum of whose distances from two fixed points (the foci) is a given(positive)constant . This is further expressed mathematically to obtain the equation of an ellipse.
An ellipse? The shape described is not an exactly describable geometric shape, (), and as far as I know there is no name for it.
An ellipse have two focal points.