Conic Sections
Conic Section: a section (or slice) through a cone.
Cones  Circle  Ellipse  Parabola  Hyperbola 

straight through  slight angle  parallel to edge of cone 
steep angle 
So all those curves are related!
Focus!
The curves can also be defined using a straight line and a point (called the directrix and focus). If you measure the distance:
the two distances will always be the same ratio.

Eccentricity
That ratio above is called the "eccentricity", so we can say that any conic section is:
"all points whose distance to the focus is equal
to
the eccentricity times the distance to the directrix"
A circle has an eccentricity of zero, so the eccentricity shows you how "uncircular" the curve is. The bigger the eccentricity, the less curved it is. 
Latus Rectum
The latus rectum (no, it is not a rude word!) runs parallel to the directrix and passes through the focus. Its length:


Here you can see the major axis and minor axis of an ellipse. There is not just one focus and directrix, but a pair of them (one each side). 
General Equation
In fact, we can make an equation that covers all these curves.
Because they are plane curves (even though cut out of the solid) we only have to deal with Cartesian ("x" and "y") Coordinates.
But these are not straight lines, so just "x" and "y" will not do ... we need to go to the next level, and have:
 x^{2} and y^{2},
 and also x (without y), y (without x),
 x and y together (xy)
 and a constant term.
There, that should do it!
And each one needs a factor (A,B,C etc) ...
So the general equation that covers all conic sections is:
And from that equation we can create equations for the circle, ellipse, parabola and hyperbola ... but that is beyond the scope of this page.