I'm going to assume that there is no "xy" term in your equations- that would be a "tilted" conic (axes NOT parallel to the x and y axes) and that is much more complicated.

Anything that has either x or y squared but not both is a parabola.

Anything that has both x and y squared and the coefficients of the squares, on the same side of the equal sign, of opposite sign, one positive and one negative, is a hyperbola.

Anything that has both x and y squared and the coefficients of the squares, on the same side of the equal sign, of the same sign, both positive or both negative, is either a circle or an ellipse. If the coefficients are the same number, then it is a circle. If the coefficients have the same sign but different values, then it is an ellipse. (Some people consider the circle a special case of an ellipse.)