In geometry, an envelope of a continuous family of differentiable curves is a curve that touches each member of that family at some point, and these points of tangency together form the whole envelope. Therefore it is the limiting curve of the intersection of contiguous members of the initial family.
Let $F(x, y, t)=0$ be a family of implicitly defined curves parametrized by variable $t.$ Any two members of this family, in general, intersect at some points. These intersections points are determined by the solution to the pair $$F(x, y, t)=0, \qquad F(x, y, t^*)=0,$$ where $t, t^*$ are the corresponding values of the parameter. As $t^*\to t$ the limiting points on the curve $F(x, y, t)=0$ are known as characteristic points. The locus of all characteristic points, in general, form another curve known as the "envelope of the initial family of curves". Envelope can be obtained by eliminating $t$ from the pair $$F=0,\qquad \dfrac{\partial F}{\partial t}=0.$$
An enlightening example is given in here with a nice animation. One can think of it as instantaneous positions of a slipping ladder. If the length of the ladder is $a,$ then $$F(x, y, \theta)=\dfrac{x}{\cos\theta}+\dfrac{y}{\sin\theta}-a=0$$ describe all members of this family of lines, where $\theta\in(0, \pi/2).$
