Semi-major axis in a binary star system The semi-major axis a here, 



what exactly does it represent? Point it out in the image below.
It is the semi major axis of the relative orbit i.e. the ellipse whose major axis is the sum of the two major axes and whose focus is in the barycentre. Essentially, this is like fixing the reference frame at the barycentre and looking at how the stars' position vectors change. It will turn out (try looking up two body problem) that this is equivalent to an object of reduced mass $\mu = \frac{m_1*m_2}{m_1+m_2}$ orbiting the one with total mass.As uhoh pointed out, this image illustrates semi major axes : https://i.stack.imgur.com/298tF.png .