What is the difference between Nucleophilic Substitution and Addition?

Put simply, in substition reactions, one thing replaces (substitutes for) another. In addition reactions, the reactants add together (so only one product).

Substitution could be A-B + C   C-B + A
Addition could be  A-B + C   A-B-C

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In nucleophilic substitution a nucleophile (such as OH-) attacks a polar C-X bond and the attacking nucleophile replaces the -X group which leaves as X-

The most common form is the substitution of OH- nucleophile for halide ions in the hydrolysis of Alkyl halides

  eg   R - Cl     +     OH-       R -OH    +   Cl-

Nucleophilic addition  involves the nucleophile attacking an electron deficient site on a polar double or triple bond. This occurs when the bond attacked is markedly polar such as the C=O bond in an aldehyde or ketone and the nucleophile can add on to the carbon atom with a partial positive charge producing a carbanion (the O atom then has  negative charge) and an electrophile can add on to the O atom subsequently. This is 1,2 nucleophilic addition.

Other groups which undergo nucleophilic addition are nitriles and imines which also have a significant charge separation across the bond.

Non polar double bonds do not undergo nucleophilic addition but those with very electronegative groups attached to one C atom may.