Conic sections are the curves which can be derived from taking slices of a “double-napped” cone. (A double-napped cone, in regular English, is two cones “nose to nose”, with the one cone balanced perfectly on the other.) “Section” here is used in a sense similar to that in medicine or science, where a sample (from a biopsy, for instance) is frozen or suffused with a hardening resin, and then extremely thin slices (“sections”) are shaved off for viewing under a microscope. If you think of the double-napped cones as being hollow, the curves we refer to as conic sections are what results when you section the cones at various angles.
A circle is a geometrical shape, and is not of much use in algebra, since the equation of a circle isn’t a function. But you may need to work with circle equations in your algebra classes.
In “primative” terms, a circle is the shape formed in the sand by driving a stick (the “center”) into the sand, putting a loop of string around the center, pulling that loop taut with another stick, and dragging that second stick through the sand at the further extent of the loop of string. The resulting figure drawn in the sand is a circle.
In algebraic terms, a circle is the set (or “locus”) of points (x, y) at some fixed distance r from some fixed point (h, k). The value of r is called the “radius” of the circle, and the point (h, k) is called the “center” of the circle.
The “general” equation of a circle is: