I'm having a little trouble understanding what makes a planetary orbit circular/explicitly elliptical. Is it simply that the initial velocity was different from the circular orbit case for a given starting position (radius from the sun) and this becomes an elliptical orbit (or more generally, an arbitrary conic section)?

Is there any other information contained in the eccentricity of an orbit?

I'm having a little trouble understanding what makes a planetary orbit explicitly elliptical. Is it simply that the initial velocity was different from the circular orbit case for a given starting position (radius from the sun) and this becomes an elliptical orbit (or more generally, an arbitrary conic section)?

Is there any other information contained in the eccentricity of an orbit?