What is exactly the third component in homogeneous coordinate system? I was reading the wiki article about homogeneous coordinates , I learned that it has it's advantages when it comes to performing affine transformation, since you can represent it only matrices. But I couldn't understand what is the additional third component compared to  Cartesian coordinates.   In the image below  we see additional 1(which  I am trying to understand)  added to convert it to  homogeneous coordinates.
The relationship between standard coordinates $(x,y)$ and homogeneous coordinates $(X,Y,Z)$ is $x = X / Z, y = Y/Z$. Homogeneous coordinates are a type of projective coordinates. All points on the line $(Z x, Z y, Z)$ are equivalent to the point $(x,y)$ in 2D (the same thing can be applied for 3D). So technically 2D points are represented as 3D lines passing though the origin, and 2D lines are represented as 3D planes passing through the origin.
In computer graphics the above interpretation is largely irrelevant, however, since homogeneous coordinates are used mostly for convenience. A translation is of the form $x' = x+t_x, y' = y+t_y$. If one uses homogeneous coordinates, then this can be achieved through $X' = 1 \cdot X + 0 \cdot Y + t_x \cdot Z$, $Y' = 0 \cdot X + 1 \cdot Y + t_y \cdot Z$, and $Z' = 0 \cdot X + 0\cdot Y + 1 \cdot Z$.  If you set $Z=1$ then you get exactly the expression for translation, and you can use the matrix induced from the 3 equations above. For directions you can set $Z=0$ so that translation would not be applied to those.
In 3D the homogeneous coordinates $W$ also plays a role in the perspective projection, since the projection matrix modifies it to hold the relevant value by which the other coordinates must be divided, and the division is typically carried out by the hardware.