orbital angular momentum
Orbital angular momentum is the quantum version of the swing a particle carries as it circulates about a centre -- the analogue of a planet's L = r cross p, now for the wave-like electron in an atom. It answers the question: how is the electron's probability cloud shaped and oriented? Its magnitude sorts orbitals into s, p, d, f shells, and its component along an axis tells you how the cloud is tilted. Crucially, both are quantized: you cannot spin an electron's orbit up by an arbitrarily small amount.
Promote the classical L = r cross p to operators and you find that the magnitude squared and one component can be known together, but the three components cannot. The eigenvalues are L^2 = l(l+1) hbar^2 with the orbital (azimuthal) quantum number l = 0, 1, 2, ..., and L_z = m_l hbar with the magnetic quantum number m_l running over the integers from -l to +l, giving 2l+1 orientations for each l. The simultaneous eigenfunctions of L^2 and L_z are the spherical harmonics Y_l^{m_l}(theta, phi), the standard shapes you meet as the angular part of every atomic orbital. Because the wavefunction must be single-valued as you go around by 2 pi, l and m_l are forced to be integers -- this is what distinguishes orbital angular momentum from spin.
Orbital angular momentum organizes the entire structure of atoms. It is the l in the spectroscopic labels 1s, 2p, 3d; it sets the shape of chemical bonds; and its quantization of orientation ('space quantization') is what the Stern-Gerlach experiment first exposed. Because it commutes with any spherically symmetric Hamiltonian, l and m_l are conserved for an isolated atom, which is why they survive as good quantum numbers and drive the selection rules governing which spectral lines an atom can emit.
For a 2p electron, l = 1, so its angular momentum magnitude is sqrt(l(l+1)) hbar = sqrt(2) hbar, and m_l can be -1, 0, or +1 -- exactly the three p-orbitals (p_x, p_y, p_z) that fan out along the axes. Note the magnitude sqrt(2) hbar is larger than the maximum projection 1 hbar, so the vector can never point purely along z: a direct fingerprint of quantization and the uncertainty principle.
The magnitude sqrt(l(l+1)) hbar always exceeds the largest projection l hbar, so L can never fully align with an axis.
The magnitude is sqrt(l(l+1)) hbar, not l hbar -- a frequent slip. That gap is not a rounding detail: it is why the angular-momentum vector can never lie exactly along the measurement axis, since doing so would pin all three components at once and violate their commutation relations.