Algebraic    Curves
Referenced from \!\(\*"\""\)\!\(\*"Algebraic"\)\!\(\*" "\)\!\(\*"Curves"\)\!\(\*"\""\) by Robert J. Walker (Princeton University Press 1950)

Example 1
An ordinary double point is called a node, it is a type of singular point, where a graph intersects itself.  It is represented by x^3-x^2+y^2 = 0.

Animate[ContourPlot[x^3-x^2+y^2==0, {x,-2,xp},{y,-2,2}, Axes->True, Frame->False, AspectRatio->Automatic, PlotRange->{{-2,1.5},{-2.5,2.5}} ,PlotPoints->100],{xp,-1.9, 2.5,.001}]

This node has two distinct tangents: x+y=0 and x-y=0, which are drawn in red.

Show[ContourPlot[x+y==0,{x,-2,2},{y,-2,2},ContourStyle->Red],ContourPlot[x-y==0,{x,-2,2},{y,-2,2},ContourStyle->Red],ContourPlot[x^3-x^2+y^2==0,{x,-2,2},{y,-2,2},Frame->False], Axes->True]

Example 2
The origin is an "isolated"  point of the real curve in this example, this isolated point is still a node, but it is called an acnode.  The difference between this curve and example1 is that the tangents of this one have imaginary values. The tangents being x+I y =0 and x-Iy = 0.

Animate[ContourPlot[x^3+x^2+y^2==0, {x,-3,xp},{y,-2,2}, Axes->True, Frame->False, AspectRatio->Automatic, PlotRange->{{-2.5,.5},{-2.5,2.5}},PlotPoints->50],{xp,-2,2,.01}]

The still shows, explicitly, the isolated point at the origin.

Show[Graphics[{Red,PointSize[.03],Point[{0,0}]},PlotRange->{{-2.5,.5},{-2.5,2.5}}],ContourPlot[x^3+x^2+y^2==0, {x,-3,2},{y,-2,2}, Axes->True, Frame->False, AspectRatio->Automatic, PlotRange->{{-2.5,.5},{-2.5,2.5}}],Axes->True, AspectRatio->Automatic]

Example 3
This figure is called a cusp,or spinode. The origin is a double point.  It is represented by the equation x^3-y^2 = 0

Animate[ContourPlot[x^3-y^2==0, {x,-1,xp},{y,-2,2}, Axes->True, Frame->False, AspectRatio->Automatic, PlotPoints->30, MaxRecursion->2,PlotRange->{{-1,1.5},{-1,1}}],{xp,-1.1,2,0.01}]

Example 4
The curve is called a Tacnode, it is also a double point like the cusp.  The tacnode is a point on a graph where the two, or possibly more, osculating circles meet at a tangent.  The tacnode below is at the origin.  The osculating part of the graph comes from the latin circulum osculans, which means \!\(\*"\""\)\!\(\*"kissing"\)\!\(\*" "\)\!\(\*"circles"\)\!\(\*"\""\).
The equation is 2x^4-3x^2y+y^2-2y^3+y^4=0.

Animate[ContourPlot[2x^4-3x^2y+y^2-2y^3+y^4==0, {x,-2,xp},{y,-.2,3},Axes->True, Frame->False,AspectRatio->Automatic, PlotRange->{{-3,3},{-1,2.5}}, PlotPoints->200],{xp,-1.9,2,.1}]

Example 5
A "ramphoid" cusp.  This graph is represented by x^4+(x^2)(x^2)-2(x^2)y-x(y^2)+y^2=0.
The name comes from the Greek ramphos "the crooked beak of birds, especially birds of prey," and that is what the curve looks like.  Also ramphoids are generally curves that have both branches one one side of the tangent, but because this cuve is actually a cusp, the tangent does not exist.

Animate[ContourPlot[x^4+(x^2)(y^2)-2(x^2)y-x(y^2)+y^2==0, {x,0,xp},{y,0,2}, Axes->True, Frame->False, AspectRatio->Automatic, PlotRange->{{0,1.5},{-0,1.5}},PlotPoints->100],{xp,0,2,.001}]

Example 6

The following is an ordinary triple point at the origin, meaning that three lines of the curve meet at one point.  It is expressed by (x^2+y^2)^2+(3(x^2))y-y^3=0.

Animate[ContourPlot[(x^2+y^2)^2+(3(x^2))y-y^3==0, {x,-2,xp},{y,-2,2}, Axes->True, Frame->False, AspectRatio->Automatic, PlotRange->{{-1.2,1.2},{-1.2,1.2}},ContourStyle->{Blue, Thickness[.001]},PlotPoints->100],{xp,-2,2,.0001}]

ContourPlot[(x^2+y^2)^2+(3(x^2))y-y^3==0, {x,-2,2},{y,-2,2}, Axes->True, Frame->False, AspectRatio->Automatic, PlotRange->{{-1.2,1.2},{-1.2,1.2}},ContourStyle->{Blue, Thickness[.001]},PlotPoints->50]

Example 7

Here the origin is of multiplicty four, basically meaning that there is four lines intersecting at one point (the origin), of this graph.  It is represted by the equation  (x^2+y^2)^3-4(x^2y^2)=0.  Although the evaluation, as shown, is done with spherical coordiantes, the evaluation of of the equation will looking something like example 6 above.

Animate[ParametricPlot[(Sin[2u])^2{Cos[u],Sin[u]},{u,0,x}, PlotRange->{{-1,1},{-1,1}}, AspectRatio->Automatic],{x,.1,2Pi,.01}]

ParametricPlot[(Sin[2u])^2{Cos[u],Sin[u]},{u,0,2Pi}, PlotRange->{{-1,1},{-1,1}}, AspectRatio->Automatic]

Example 8

Here something different happens, the origin is a lot more complex than anything above because it has a triple tangent and two simple tangents.

Animate[ContourPlot[x^6-x^2y^3-y^5==0,{x,-2,xp},{y,-2,2}, Axes->True, Frame->False, AspectRatio->Automatic, PlotRange->{{-2,2},{-2,2}}, PlotPoints->100],{xp,-2,2,.0001}]