Molecular nitrogen is a spin singlet (and so diamagnetic) and molecular oxygen is a spin triplet (and so paramagnetic).
The figure below taken from Atkins' Physical Chemistry (Figure 10.33 of the ninth edition) illustrates the comparative molecular orbital electronic structure. The key difference is that the two valence electrons in oxygen are in two degenerate pi_g orbitals. Hund's rule coupling then causes the ground state to be a spin triplet. There is a large Curie paramagnetism associated with that.
Update. (August 2025).
There is a helpful discussion on the Wikipedia page for singlet oxgyen, containing the diagram below.
This indicates that the Hund's coupling is of order 1 eV.
In Modern Quantum Chemistry, Szabo and Ostlund state (p.220-1) that "The first brilliant success of molecular orbital theory was the explanation of why O2, with an even number of electrons, does not have all its electrons paired."
A nice discussion of the relevant two-site two-orbital Hubbard model is here.
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