Initial remark. Suppose someone gives you a continuous function of two variables, and asks you to calculate the Taylor series in $(0,0)$. If we happen to notice that the first derivatives vanish at the origin, we will call it a critical point.
The idea of the tangent cone is that we are doing exactly this: taking the Taylor series of the simplest continuous function: a polynomial. And if the first derivatives vanish, we call $(0,0)$ a singular point.
We have a curve $X=V(f)\subset \mathbb A^2$ passing through the origin.
Even if the origin is a nonsingular point of the curve, you do have a tangent cone at the origin: it is also called the tangent line! (and this is precisely encoded in the corresponding first terms of the "Taylor series" of $f$).
However, the tangent cone is constructed starting from the leading form of $f\in k[x,y]$, which is the homogeneous form $\tilde f$ of smallest degree appearing in the decomposition of $f$; it is again in two variables. For instance, the leading form of $f=2x+y-8y^2x$ is $\tilde f=2x+y$, the term of smallest degree. The tangent cone is the zero set $V(\tilde f)$. So the origin is nonsingular if and only if $f$ has leading form of degree one. In that case, $V(\tilde f)$ is exactly the tangent line.
In general, the leading form will be a product of linear factors, each appearing with some exponent. So the scheme $V(\tilde f)$ is a union of lines, where some of them are possibly nonreduced.
If $f$ starts, say, with degree $2$, then for instance $(0,0)$ will be a node in case the (degree $2$) leading form is a product of two distinct linear forms, i.e. something of the kind $$(ax+by)\cdot (cx+dy).$$ If, instead, the leading form has the shape $(ax+by)^2$, something else happens (example: the cusp $f=y^2-x^3$, whose tangent cone which is a double line).
Reference. All this is beautifully explained in Mumford's The red book of varieties and schemes.