Markovnikov's rule, and when it does not hold

Adding HX across an alkene puts the hydrogen on the carbon that already has more of them, and the X on the other. The memorable phrasing hides the actual reason, which is that the reaction goes through whichever carbocation is more stable.

Propene and water under acid. The OH ends up on the middle carbon, because that is where the more stable cation formed.

Why it is really about the cation

The proton adds first, and it can land on either carbon. Whichever way it lands leaves a positive charge on the other one. Alkyl groups stabilise that charge, so the proton effectively lands wherever leaves the more substituted carbocation — which is the carbon with fewer alkyl groups, the one already carrying more hydrogens.

Stating it as the stability of the intermediate rather than as a counting rule is worth doing, because it is the version that keeps working when the counting version stops. It also explains the rearrangements: if a hydride or alkyl shift gives a better cation, it happens, and the product is not the one the counting rule predicts.

The anti-Markovnikov cases

Hydroboration puts boron on the less substituted carbon and hydrogen on the more substituted one, which is the opposite placement. Nothing about the rule is being violated: there is no carbocation, because boron and hydrogen add across the bond together in one step, and sterics decide where the boron goes.

Radical addition of HBr in the presence of peroxides does the same thing by a different route, through a bromine radical adding to give the more stable carbon radical.

So the useful statement is not that Markovnikov's rule has exceptions. It is that the rule describes what happens when the reaction goes through a cation, and reactions that do not go through one are not exceptions to it at all.

Where this stops being simple

The rule predicts which regiochemistry dominates, not that the other product is absent. Both usually form.
It says nothing about stereochemistry. A new stereocentre made through a flat carbocation forms as a mixture.

See it happen

Acid-catalysed hydration, drawn

The protonation, the cation and the capture, step by step.

Nucleophile and electrophile

The alkene is the nucleophile here, which is worth seeing explicitly.

Try an addition

Draw an alkene and a reagent and see which outcome the engine ranks first, and on what evidence.

Other concepts

What is the difference between SN1 and SN2?

Both replace a leaving group with a nucleophile, and they differ in whether the bond breaks before the new one forms. SN2 does it in one motion and turns the carbon inside out; SN1 lets the leaving group go first, makes a carbocation, and whatever attacks it can attack from either face.