We consider motion as the movement of a body. For example, when you move from your initial position, you consider your movement. Now we only focus on the motion without considering the cause of motion, the force. If a particle moves in a two dimensional coordinate system i.e. in a plane, both of its coordinates change and we call that motion as the two dimensional motion. And if the particle movies in space i.e. in a three dimensional coordinate system, the motion of the particle is three dimensional.
In everyday life you might think the distance and displacement are the same things but they are totally different in physics. Distance is a scalar quantity and displacement is a vector quantity. Being scalar distance does not have any particular direction associated with it but displacement has direction.
In Figure 1 a particle moves from the initial position $s_1$ to the final position $s_2$ along the curved path. In this case the distance represents the whole path of motion but the displacement only depends on the initial and final positions. Therefore, the displacement is the straight line joining the two positions which has direction towards the final position from the initial position (as the particle reached the final position).
In other words the displacement does not depend on the details of motion of the particle. If you consider this information in a coordinate system, if the initial position vector of point $s_1$ is $\vec r_1$ and the final position vector of point $s_2$ is $\vec r_2$, the displacement is $\Delta \vec r$ = $\vec r_2 - \vec r_1$. If you don't know what a position vector is, it is simply a vector representing a particular position in a coordinate system.