The right-hand rule is a convention in vector math. It helps you remember direction when vectors get cross multiplied. There is another rule called the Right-hand fist rule that is used for magnetic fields and things that rotate.
If you have two vectors that you want to cross multiply, you can figure out the direction of the vector that comes out by pointing your thumb in the direction of the first vector and your pointer in the direction of the second vector. Your middle finger will point the direction of the cross product.
Remember that when you change the order that vectors get cross multiplied, the result goes in the opposite direction. So it's important to make sure that you go in the order of .