Imagine you have a toy car that moves back and forth. Now, if you add a toy spinning top on top of the car, it starts moving in a circular motion. This spinning top represents the spin of a particle, like an electron or a proton. The direction in which it spins is called the spin direction.
Now, imagine that you want to measure the total amount of spinning energy of the toy car and the spinning top combined. To do this, you need to know both their velocities and their spinning direction. This total amount of spinning energy is called the Pauli-Lubanski pseudovector.
It's like putting together all the information we have about the car and the spinning top and coming up with a single number to represent how much they are spinning. This is important in physics because measuring the spin of a particle helps us understand its properties and behavior, and the Pauli-Lubanski pseudovector makes this easier.