There are two points: the foci.
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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.
The two fixed points are the foci but these do not define the shape of the ellipse. You also need to know the eccentricity.
An ellipse is a shape on which the sum of the distances from every point to two points inside called the foci (focuses) is always the same number. A circle is an ellipse with both foci (focuses) at the same point.