a x b = |a||b| sin t nwhere n is a unit vector perpendicular to the plane in which a and b lie. (I've shown n as part of the vector a x b.) We choose the direction of n so that a, b and n can fit along the thumb, first finger and second finger of your right hand. (Check this with your hand and my drawing.)
Notice that this means that the direction of b x a is given by - n so a x b = - b x a.
The multiplication sign used to show the vector product is called 'cross'. It is also sometimes written as a little upside down v like a chinese hat.