Markovnikov's rule
In chemistry, Markovnikov's rule or Markownikoff's rule is an observation based on Zaitsev's rule. It was formulated by the Russian chemist Vladimir Vasilevich Markovnikov  in 1870  . In chemical reactions found particularly in organic chemistry, the rule states that with the addition of H-X to an alkene, the acid hydrogen (H) becomes attached to the carbon with the greatest number of hydrogens, and the halide (X) group becomes attached to the carbon with the fewest number.
The same is true when an alkene reacts with water in an addition reaction to form alcohol. The hydroxyl group (OH) bonds to the carbon that has the greater number of carbon-carbon bonds, while the hydrogen bonds to the carbon on the other end of the double bond, that has more carbon-hydrogen bonds.
The chemical basis for Markovnikov's Rule is the formation of the most stable carbocation during the addition process. The addition of the hydrogen to one carbon atom in the alkene creates a positive charge on the other carbon, forming a carbocation intermediate. The more substituted the carbocation (the more bonds it has to carbon or to electron-donating substituents) the more stable it is, due to induction and hyperconjugation. The major product of the addition reaction will be the one formed from the more stable intermediate. Therefore, the major product of the addition of HX (where X is some atom more electronegative than H) to an alkene has the hydrogen atom in the less substituted position and X in the more substituted position. It is important to note, however, that the other less substituted, less stable carbocation will still be formed to some degree, and will proceed to form the minor product with the opposite attachment of X.
The rule may be summarized as "the rich get richer and the poor get poorer" or "them that has gets": a carbon rich in substituents will gain more substituents and the carbon with more hydrogens attached will get the hydrogen in many organic addition reactions.

# Historic context
One the organic reactions Markovnikov based his rule on (first performed in 1865) was that of hydrogen iodide with vinyl bromide. In another manifestation of his rule he observed that the halogen atom added to the carbon atom already carrying an halogen atom  Geminal halide hydrolysis of the initial reaction product with moist or hydrogenated (rarely used) silver oxide to ethanal proved the 1,1 substitution pattern.
It has been observed  that the original 1869 Markovnikov publication was sloppy and that he did not do much experimental work himself. The rule itself appeared only as a four-page footnote in a 26-page article and summarized by Markovnikov as "Them as has, gits.",  which may also explain why his rule took 60 years to catch on.

# Anti Markovnikov rule
Mechanisms which avoid the carbocation intermediate may react through other mechanisms that are regioselective, against what Markovnikov's rule predicts, such as free radical addition. Such reactions are said to be anti-Markovnikov, since the halogen adds to the less substituted carbon.This reaction is exactly opposite of Markovnikov reaction,and hence the name. Again, like the positive charge, the radical is most stable when in the more substituted position.
Anti-Markovnikov behaviour extends to other chemical reactions than just additions to  alkenes. One Anti-Markovnikov manifestation is observed in hydration of phenylacetylene that, gold-catalyzed, gives regular acetophenone but with a special ruthenium catalyst  the other regioisomer 2-phenylacetaldehyde  :
Anti-Markovnikov behaviour can also manifest itself in certain rearrangement reactions. In a titanium(IV) chloride catalyzed  formal nucleophilic substitution at enantiopure 1 in the scheme below, two racemic products are formed 2a and 2b  :
This product distribution can be rationalized by assuming that loss of the hydroxy group in 1 gives the tertiary carbocation A which rearranges to the seemingly less attractive secondary carbocation B. Chlorine can approach this center from two faces leading to the observed mixture of isomers.