For some background on circles, parabolas, ellipses and hyperbolas you can look up Conic Sections on the Web.
The link http://www.vhcc.edu/pc2fall9/frames%20pages/class_notes.htm takes you to my Class Notes for Precalculus II. You might find those pages helpful.
A quick summary:
The equation y = a x^2 is the basic parabola opening vertically, while x = a y^2 is a parabola opening horizontally. In each case the vertex is the origin.
If the vertex is the point (h, k) then we replace x and y respectively by x - h and y - k, which shifts the graph in the horizontal and vertical directions so that the origin moves to the point (h, k).
The equation x^2 / a^2 + y^2 / b^2 = 1 is an ellipse which will fit into the rectangular box defined by the lines x = a, x = -a, y = b and y = -b. This ellipse is centered at the origin.
If the center of the ellipse is the point (h, k) then we replace x and y respectively by x - h and y - k, which shifts the graph in the horizontal and vertical directions so that the center moves to the point (h, k).
The equations x^2 / a^2 - y^2 / b^2 = 1 and the equation - x^2 / a^2 + y^2 / b^2 = 1 are hyperbolas which can be sketched using the rectangular box defined by the lines x = a, x = -a, y = b and y = -b. In either case the hyperbola will be asymptotic to 'diagonal' lines of the box (imagine extending the diagonals of the recangle indefinitely in both directions). The equation x^2 / a^2 - y^2 / b^2 = 1 has vertices at the points (a, 0) and (-a, 0), the points where the x axis intersects the box; it opens to the right and to the left. The equation -x^2 / a^2 + y^2 / b^2 = 1 has vertices at the points (0, b) and (0, -b), the points where the y axis intersects the box; this hyperbola opens up and down.
The two asymptotes of either of these hyperbolas meet at the origin. If the asymptotes meet at the point (h, k) then we replace x and y respectively by x - h and y - k, which shifts the graph in the horizontal and vertical directions so that the intersection of the asymptotes moves to the point (h, k).