Here I have an example which I found. Can someone help me to understand what's happening here? The following are my concerns:
1) What do we need to do co-ordinate transformation?
2) How does the initial polynomial help us to see whether the singularity is ordinary or not?
3) How can we read tangents directly from the initial polynomial?
To answer your first question, if we want to analyze a point $(x_0,y_0)$ of a curve $f(x,y)=0$ (let's just assume it's affine), we want to first see $f$ as a polynomial in the variables $x-x_0$ and $y-y_0$. This is done by writing $x=x-x_0+x_0$ and $y=y-y_0+y_0$. So, for example, if we have the curve $x^2+y^2-1=0$ and we want to analyze the point $(1,0)$, we just write $$x^2+y^2-1=(x-1+1)^2+y^2-1=(x-1)^2+2(x-1)+y^2.$$
Step 2: To analyze a singularity, it helps to look at the tangent cone of your curve. If we are centered at $(x_0,y_0)$ as above, then this is just the vanishing set of the homogeneous part of your polynomial of lowest degree. So in the case of $x^2+y^2-1$, if we are centered at $(1,0)$ then the tangent cone corresponds to $x-1=0$. Notice that if a point is non-singular, then the tangent cone is just the tangent line.
Let's look at the example $y^2-x^2-x^3$ at $(0,0)$. Here we see that the tangent cone is $y^2-x^2=0$, that is $(y-x)(y+x)=0$. This gives us the union of two "tangent" lines $y=x$ and $y=-x$. This shows that the singularity at $(0,0)$ is a node. This also answers your question 3).