Ellipses have two foci. The sum of the distances from each of the foci to a point on the ellipse is always the same. As the two foci move closer and closer, the ellipse becomes less and less eccentric -- closer to a circle. When the foci become the same point, the ellipse becomes a circle. Both circles and ellipses are created when a plane intersects a cone. Circles are made when the plane intersects the cone at right angles to the cone's central axis. The ellipse is made when the plane intersects the cone at an angle other than a right angle. For example, the shapes of the planets' orbits around the sun are ellipses.
Parabolas are described by a focus and a line called the directrix. Each point of a parabola is an equal distance from the focus and the directrix. Parabolas are roughly U-shaped curves in which the ends of the U are infinitely long. The focus is inside the parabola, and the directrix is outside the parabola. The interesting thing about parabolas is the way parallel lines coming into the inner curve of a parabola bounce off the curve in such a way that they concentrate at the focus. This property is exploited in parabolic satellite dishes. Parabolas are models of many natural phenomena. When a cannon ball is fired, for example, it traces out a parabola when returning to Earth.
Hyperbolas look like two parabolas that are nose to nose, although they are different from parabolas. The two branches of the hyperbola have foci inside the two branches of the curve, and there is a directrix-like line. The big difference is that a hyperbola consists of points in which the differences between the distances to the foci are the same. Hyperbolas describe the shape of the surface of telescope mirrors or the curve of rainbows.
Describing conic section style curves is part of the mathematical heritage we inherited from the Greeks. Modern algebraic techniques eliminate the need for foci in describing these curves. For example, there is a single equation that describes all of these curves: Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0, where A, B, C, D, E and F are constants. When B^2 - 4AC is less than zero, the formula describes circles and ellipses. When B^2 - 4AC = 0, the formula describes parabolas. When B^2 - 4AC is greater than zero, the formula describes hyperbolas.