A function $f$ is not differentiable at $a$ if it has no tangent at $a$.

Given a curve $C$ and a point $P$ on $C$, what does it mean to say that a line $L$ is tangent to $C$ at $P$?

Well, the first requirement is that $L$ passes through $P$. But, there are many different lines that pass through $P$. So, for instance, if $C$ is the graph of $y=|x|$ and if $P=(0,0)$ then the line $y=0$ passes through $P$, and the line $x=0$ passes through $P$, and the line $y=x$ passes through $P$, and the line $y=.01x$ passes through $P$.

What's so special about tangent lines?

The additional special property of the tangent line is not just that it passes through $P$, but that at points of the curve $C$ nearby $P$ the tangent line $L$ is a very good approximation of $C$. Intuitively, what this means is that if you took a microscope and looked at the region around $P$ through that microscope, the curve $C$ and the line $L$ would look very much like each other near $P$, but perhaps $C$ curves a bit away from $L$ somewhat. But then you have to repeat this intuition with more and more powerful microscopes. If you take a very powerful microscope, and looked at the region around $P$ through that microscope, the curve $C$ and the line $L$ would look very very much like each other near $P$, but perhaps $C$ curves a very little bit away from $L$.

Now let's apply this intuition to the case of $y=|x|$ and $P=(0,0)$. No matter how powerful of a microscope you take, when you look at the region around $P$ the graph of $y=|x|$ and the line $y=0$ look nothing like each other, other than the fact that they both contain $P$. So no, the line $y=0$ is not a tangent line to $y=|x|$, and in fact there does not exist any line is tangent to $y=|x|$ at $P$.

Of course, all these notions of very very close seem rather intuitive. But that's the point of calculus: it makes those kind of notions logically precise, using the concept of a limit. And when you make the definition of tangent line precise, for example as in the answer of @Griboullis, then you will also see that $y=|x|$ has no tangent line at $(0,0)$.