A conic in n dimensions is the zero set of a degree 2 polynomial. For example xy-1=0 in 2 dimensions has a hyperbola, a kind of conic. All conics can be realized as the intersection of a plane with a cone in one higher dimension, hence the name.
A union of conics is the zero set of the product of many degree 2 polynomials, in 3 dimensions for example you could have
(x^2+y^2-1)(x^2-y)(x^2+y^2-z^2)=0, the union of 3 conics: a circular cylinder, a parabolic cylinder, and a cone. A product as such is literally the set theoretical union of conics.
