The simplest, most commonly taught picture of a chemical bond treats it as a pair of electrons localised between two specific atoms — this is the essence of valence bond theory, and it's a genuinely useful, intuitive way to think about most simple bonds. Molecular orbital theory offers a fundamentally different, more mathematically rigorous picture: rather than treating electrons as belonging to a bond between two specific atoms, it treats them as occupying orbitals that extend across the molecule as a whole, built by combining the original atomic orbitals of every contributing atom into new, molecule-wide orbitals in which the electrons genuinely reside.
Combining atomic orbitals produces molecule-wide ones
Molecular orbital theory starts from the atomic orbitals of the individual atoms making up a molecule and mathematically combines them to produce a new set of molecular orbitals, each one spread out across the molecule rather than confined to a single bond between two specific atoms. Combining two atomic orbitals produces two new molecular orbitals: a lower-energy bonding orbital, where the combination reinforces electron density between the two nuclei in a way that holds them together, and a higher-energy antibonding orbital, where the combination instead reduces electron density between the nuclei in a way that would, if filled, work against the atoms staying bonded. Electrons then fill these new molecular orbitals from lowest energy to highest, following essentially the same filling rules used for atomic orbitals, and it's the specific balance between how many electrons end up in bonding versus antibonding orbitals that determines a molecule's actual bond strength and stability.
What the delocalised picture predicts that the simpler picture can't
Because molecular orbital theory treats electrons as genuinely delocalised across the whole molecule rather than confined to individual two-atom bonds, it can correctly predict several real molecular properties that the simpler, more localised valence bond picture struggles to explain cleanly — including molecular oxygen's genuine magnetic behaviour, a real physical property arising from unpaired electrons that molecular orbital theory correctly predicts but that the simpler electron-pair picture doesn't straightforwardly account for, and the behaviour of molecules with extended systems of alternating bonds, where the electrons involved are demonstrably not confined to any single specific bond location the simpler model would place them at. This is why molecular orbital theory, despite being mathematically more demanding to actually work through than the simpler valence bond picture, has become the standard framework in more advanced and computational chemistry, even though the simpler picture remains genuinely useful as a quick, intuitive first approximation for many everyday bonding questions.
What we're still unsure about
The core mathematical framework of molecular orbital theory, and its demonstrated ability to correctly predict properties like molecular oxygen's magnetic behaviour that the simpler valence bond picture misses, are well-established, thoroughly verified quantum chemistry. What remains a genuinely active area of computational chemistry research is how to make molecular orbital calculations for very large, complex molecules — proteins, or large industrial catalysts, for instance — both accurate and computationally practical to actually run, since the calculations involved grow dramatically more demanding as a molecule's size and complexity increase, and chemists are still actively developing new computational approximation methods that trade off some precision for the ability to model much larger real systems within a realistic amount of computing time.
This sits inside Quantum Chemistry & Molecular Orbitals, one of seven topics in Physical Chemistry, one of six domains in Chemistry, one of seventeen subjects the app can quiz you on.