The Right hand rule is a useful mnemonic for visualizing the direction of a magnetic force as given by the Lorentz force law. The diagrams above are two of the forms used to visualize the force on a moving positive charge. The force is in the opposite direction for a negative charge moving in the direction shown. One fact to keep in mind is that the magnetic force is perpendicular to both the magnetic field and the charge velocity, but that leaves two possibilities. The right hand rule just helps you pin down which of the two directions applies.
For applications to current-carrying wires, the conventional electric current direction can be substituted for the charge velocity v in the above diagram.