Lewis acids and bases are electron acceptors and donors, respectively. Electrophiles are Lewis acids, and Nucleophiles are Lewis bases.
Lewis acids and bases include Bronsted acids and bases, which in turn include Arrhenius acids and bases. That’s potentially a bit confusing to read, so please enjoy this colorful diagram:
Notice that the Lewis circle encompasses both Bronsted-Lowry and Arrhenius and that Bronsted encompasses only Arrhenius.
Pro-tip: any time a proton is exchanged, electrons made it happen! That’s true for everything else in chemistry, btw—electrons are kind of a big deal.
What exactly is the difference between an Arrhenius acid vs a Bronsted acid? By definition, Arrhenius acids and bases are restricted to water while Bronsted acids and bases can be anything else—the only requirement is a proton exchange.
Since Lewis acids are the broadest definition of acids, I'll be using that definition for the following examples of acids and bases.
Remember how I was saying that Bronsted acids are Lewis acids? Let’s check out the reaction between acetic acid and hydroxide. The curved arrows, as always, indicate the flow of electrons:
The blue oxygen of the hydroxide anion donates a lone pair to grab the hydrogen. Hydrogen can only make one bond, so the green oxygen accepts the electron pair that was in its bond to hydrogen. The Bronsted acid is acting as a Lewis acid! And the Bronsted base is acting like a Lewis base!
Lewis acids are electron acceptors, so they’re called electrophiles; Lewis bases are electron donors, and they’re called nucleophiles. Many, but not all, Lewis acids have empty p-orbitals. Nucleophiles can attack that electrophilic empty orbital to form an adduct (sometimes called a complex ion).
The most common Lewis Acids are the "A, B, Cs" = Aluminum (AlX3), Boron (BX3), and Carbocations (C+)
In this image, two reactions are happening. Ammonia's lone pair (non-bonding electrons) on the nitrogen is attacking the empty orbitals of the central atoms of BF3 and AlCl3. Notice that a new bond is formed between nitrogen and boron or nitrogen and aluminum. Since nitrogen donated its electrons, it’s a Lewis base (nucleophile); since boron and aluminum accepted electrons, it’s a Lewis acid (electrophile).
Molecules without empty orbitals can act as Lewis acids as well. How might that happen? Polar molecules like acetone have an electrophilic portion that loves to act as an electron-pair acceptor. Let’s use hydroxide as the electron-pair donor.
Lewis acids are often used to lower the activation energy in electrophilic aromatic substitution reactions like halogenation. Benzene has a special kind of stability called aromaticity, so a Lewis acid catalyst needs to be present:
There’s a lot going on in this reaction, but what’s important for us right now is to see how the bromine in blue is making a bond to the empty orbital of the iron to form a complex. That makes the resulting FeBr4 a good leaving group, so the aromatic molecule can now easily attack the electrophilic Br.