Here's a place to see the normal cone side-by-side with other familiar constructions, that I learned from Fulton's "Intersection Theory".

Start with the space $Y \times {\mathbb P}^1$, thought of as a trivial family over ${\mathbb P}^1$. Blow up the subscheme $X \times \infty$. Now we still have a flat family over ${\mathbb P}^1$, in which all the fibers except the one over $\infty$ are still copies of $Y$. The fiber over $\infty$ is reducible: one piece $Z_1$ is the blowup of $X$ along $Y$, and the other is the projective completion $Z_2$ of the normal cone to $Y$ inside $X$. These intersect along the projectivization of the normal cone, which appears in $Z_1$ as the exceptional divisor, and in $Z_2$ as the stuff added in projective completion.

(An example, for people who like polytopal pictures of toric varieties: let $Y$ be ${\mathbb P}^2$, pictured as a triangle, and $X$ be a point, pictured as a vertex. Then $Y \times {\mathbb P}^1$ is pictured as a triangular prism, and its blowup by cutting a corner off that prism. The fiber over $\infty$ is then pictured as cutting that triangle into a trapezoid $Z_1$ union a small triangle $Z_2$, glued along an interval, matching the decomposition above.)

So, if perhaps you don't like degenerating $X$ (which may be complete) to something noncomplete, you can complete it by including $Z_1$. The normal cone is $Z_2 \setminus Z_1$.

Two other comments. On $Y \times {\mathbb P}^1$ there is a circle action, dilating the ${\mathbb P}^1$. It acts trivially on the $0$ and $\infty$ fibers, moving the rest around. When we blow up $X \times \infty$, the circle action on the new $\infty$ fiber $Z_1 \cup Z_2$ is nontrivial on $Z_2$; it is the dilation of the cone fibers. This is one place to lay blame for the existence of this "cone" structure.

Finally, there's a conformally equivalent way to think about the $Y$ to $Z_1 \cup Z_2$ degeneration, at least when $X$ and $Y$ are smooth. Pick a small tubular neighborhood (nonalgebraic!) around $X$, with boundary $S$, a sphere bundle over $X$. Let the metric on $Y$ get very looong nearby $S$, in directions passing through $S$. You might say that $X$ is falling into a black hole, with $S$ the Schwarzschild boundary. In the limit, it gets infinitely long, and $X$ has bubbled off into its own universe.