Nucleophilic addition is essentially the opposite of elimination - shown in general form below;

The two main types of unsaturated systems that will undergo this are alkenes and carbonyls. These two systems are very different, so their examples will be kept separate, but a general pattern should become evident. The other type of addition that occurs (also onto unsaturated systems) is electrophilic addition, which has a section of its own, and will not be covered here.

Alkenes

Alkenes which have attached to them electron withdrawing groups (abbreviated 'EWGs' - examples below) will be susceptible to nucleophilic addition.

An alkene is a naturally electron rich system, so will inherently appeal to electrophiles rather than nucleophiles - however, with a suitable degree of electron deficiency caused by EWGs, they can be attacked by nucleophiles. The typical EWG will act by pulling electrons out of the double bond - either inductively (electronegative atoms), or mesomerically (by resonance). The crucial point is that the EWG has to withdraw π electron density from the double bond - usually by resonance into π orbitals. A couple of EWG examples;

Here is a less general example of a nucleophilic addition occurring on an alkene;

Points to note:

i. A carbanion is produced by the attack of the nucleophile.

ii. Some sort of electrophile (often H⁺) must be present to react with the carbanion.

The important point of (i) is that the nucleophile will attack the end of the alkene so as to leave the carbanion where it is most stabilised by electron withdrawing substituents. Another way of putting this is to say that the EWGs will pull electrons along the double bond towards them, leaving the end furthest away with least electrons, and therefore δ⁺ which is most attractive to the incoming nucleophile.

A real example of nucleophilic addition to alkenes is the Michael Reaction, where a carbanion is the nucleophile;

Note that the carbanion is stabilised by resonance as an enolate - this is a very common way of stabilising carbanions. Both resonance structures have been drawn, but it has also been made clear that it is the resonance structure with the negative charge on the carbon that reacts.
The above reaction is only an example of a Michael Reaction - the name applies to all nucleophilic addition reactions where a carbanion is the nucleophile. It is useful in synthesis because a carbon-carbon bond is formed.