What is exactly the third component in homogeneous coordinate system?
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.