OK, that's what we use for orbital angular momentum, like the Earth going around the sun. The cross product gives you a vector that's perpendicular to the plane of the orbit. In QM, the vector \vec L is quantized, both in magnitude and in direction. The magnitude is quantized according to
| \vec L | = L = \sqrt{l(l+1)} \hbar
where l is an integer 0, 1, 2, 3... Furthermore, the component of \vec L along any direction is quantized. Usually we talk about the z-direction but it can actually be any direction. After you've chosen the value of l, then
L_z = m_l \hbar
where m_l can have values ranging from -l to +l in steps of 1. For example, if l = 2, then the possible values of m_l are -2, -1, 0, +1, +2.
Something like the Earth also has spin angular momentum, from spinning around its own axis. We can describe this with a vector \vec S that points along the axis of rotation. Even though particles like electrons actually aren't little tiny spinning balls, they still have intrinsic angular momentum which we often call "spin," and we use the vector \vec S for it.
The rules for quantizing \vec S are similar to the rules for quantizing \vec L:
| \vec S | = S = \sqrt{s(s+1)} \hbar
S_z = m_s \hbar
where m_s can have values ranging from -s to +s in steps of 1. The differences from orbital angular momentum are:
1. l must be an integer, but s can be either integer or half-integer.
2. For a particular particle (e.g. electron) l and/or m_l can change when its "orbit" changes, but s is always the same for a particular kind of particle. For example, all electrons have s = 1/2, and so they must have m_s = -1/2 ("spin down") or m_s = +1/2 ("spin up").
However, m_s can change. With the right circumstances, we can "flip" an electron's spin from m_s = -1/2 to m_s = +1/2 or vice versa.
