The semi-major and semi-minor axes of an ellipse are radii of the ellipse (lines from the center to the ellipse). The semi-major axis is the longest radius and the semi-minor axis the shortest. If they are equal in length then the ellipse is a circle. Drag any orange dot in the figure above until this is the case.
Each axis always meets the other at the center at right angles.
The focus points always lie on the major (longest) axis, spaced equally each side of the center. So one will always lie on the semi-major axis. See Foci (focus points) of an ellipse. In the figure above, reshape the ellipse and note the behavior of the two black focus points.
The semi-major and semi-minor axes are half the length of the major and minor axis. To calculate their lengths, use one of the formulae at Major / Minor Axis of an ellipse and divide by two.
Some find the names 'semi-major / semi-minor axis' cumbersome and confusing. Typically, an axis passes all the way through an object and is an axis of symmetry. In the semi case that is not so. Also, they are usually used as a length (see Area of an ellipse) rather than a line segment. For these reasons, some prefer to call them the major radius and minor radius of the ellipse.