According to the Lewis definition, Lewis acids are substances that accept electron pairs - electron acceptors, while Lewis bases are substance that donate electons - electron donors.
A classic example of a Lewis acid-base reaction takes place between ammonia (NH_3) and boron trifluoride (BF_3).
Since boron (B) is in group 13, it only has 3 valence electrons, which means it can only form three bonds. B bonds with three F atoms and adds 3 more electrons to its outermost shell, for a total of 6 electrons (3 it owns and 3 come from the covalent bonds it has with F).
Notice that it still needs 2 more electrons to complete its octet, i.e. to have 8 electrons in its outermost shell, therefore can accept a pair of electrons. This is where NH_3 comes into play.
N is in group 15, which means it has 5 valence electrons and can form three bonds (this is because out of the five valence electrons, only 3 are unpaired) - it can therefore donate a pair of electrons.
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Here's what happens during their reaction. Since it is bonded to three other atoms, B is sp^2 hybridized. This means that one of its three 2p-orbitals (2p_z) is empty. The lone pair of electrons on N is picked up by B's empty p-orbital and a covalent bond is formed between the two molecules.
NH_3 acted a Lewis base - it donated a pair of electrons, while BF_3 acted as a Lewis acid - it accepted a pair of electrons.
