What Does Homogenisation Of An Equation Actually Mean? For example, if we have a conic:

$$ax^2 + 2hxy + by^2 + 2gx + 2fy + c = 0$$

What does homogenising this equation with another line (say $ax + by + c = 0$ ) actually mean? As in, what are the graphical consequences of this? I've been told that it represents a pair of straight lines passing through the origin and the points of intersection (of the conic and the line), but how did this come about?

Update: I'm sorry for the lack of context, but I've just started algebraic geometry and I like understanding what things actually mean instead of mugging up a formula.
A function $f:\mathbb{R}^n \to \mathbb{R}$ is homogeneous of degree $r$ when it has the following property:

$$f(\lambda x_1,\lambda x_2,\ldots,\lambda x_n) = \lambda^r f(x_1,x_2,\ldots,x_n)$$

Examples of such functions include:


Linear functions, they are of degree 1. If you scale the graph of the function by a factor $\lambda$, you still get the same graph, except that all points have coordinates scaled up by the factor $\lambda$.
The monomials in one variable: $f:\mathbb{R} \to \mathbb{R}: x \mapsto x^n$ is homogeneous of degree $n$, scale $x$ by a factor $\lambda$ and the function will scale by a factor $\lambda^n$.
Any polynomial (function) in $n$ variables such that each term appearing in the polynomial is of degree $k$ is homogeneous of degree $k$. For instance $x^2 y + z^3$ is homogeneous of degree $3$.


The importance of homogeneity is the scale invariance of the functions. Which implies that the graphs of the functions will be scale invariant. Indeed, imagine a homogeneous function is used to define a geometrical object implicitly:

$$f(x_1,x_2,\ldots,x_n)=0$$

meaning all points with coordinates $(x_1,\ldots,x_n)$ that satisfy this equation will belong to the geometrical figure defined by $f$. If $f$ is homogeneous, it immediately follows that any multiple of these coordinates also satisfies the equations. In other words, any point that satisfies the equation immediately implies the entire ray going through that point and the origin of the space belong to the geometrical object. 

Homogenizing an implicit polynomial equation means adding an extra variable $z$ and multiply any term by $z^k$ with $k$ such that the resulting polynomial is homogeneous. Of course, since any $z$-multiple of the polynomial will also be homogeneous, you choose the resulting homogeneous polynomial with smallest possible degree.

In your example, this would become

$$ax^2 + 2hxy + by^2 + 2gxz + 2fyz + cz^2 = 0$$

If you take the intersection of the geometrical figure defined by above equation with the plane $z=1$, you get back the original figure. If you choose another plane parallel with the $z=1$ plane, you get a scaled up or scaled down version of the figure.
Homogeneous equation should have [all] terms of equal powers..
For Example: $$x^2 + y^2 + 2xy = 0$$

the degree of $x^2$, $y^2$, and $xy$ is 2, hence the equation is the homogeneous equation of 2 degree.