Solidification proceeds as the molten metal ahead the solid-liquid interface becomes supercooled. As heat is removed from the interface regions, either by convection in the molten metal or by conduction through the solid and/or mold, solidification can proceed by thermal supercooling of the melt. This behavior is a predominant mechanism in pure metals. In addition, alloys can freeze by constitutive supercooling, where rejection of solute by the solid into the melt will lower the melting point in the boundary layer of the remaining liquid, thus widening the freezing range of the alloy. So, constitutive supercooling arises because of the segregation of alloying elements ahead of the solid-liquid front. Constitutive supercooling occurs only in alloys, and the thermal supercooling is operative in both pure metals and alloys.

Quotes from books of K.A. Walsh “Beryllium Chemistry and Processing”, F.C. Campbell “Elements of Metallurgy and Engineering Alloys”.

Solidification proceeds as the molten metal ahead the solid-liquid interface becomes supercooled. As heat is removed from the interface regions, either by convection in the molten metal or by conduction through the solid and/or mold, solidification can proceed by thermal supercooling of the melt. This behavior is a predominant mechanism in pure metals. In addition, alloys can freeze by constitutive supercooling, where rejection of solute by the solid into the melt will lower the melting point in the boundary layer of the remaining liquid, thus widening the freezing range of the alloy. So, constitutive supercooling arises because of the segregation of alloying elements ahead of the solid-liquid front. Constitutive supercooling occurs only in alloys, and the thermal supercooling is operative in both pure metals and alloys.

Quotes from books of K.A. Walsh “Beryllium Chemistry and Processing”, F.C. Campbell “Elements of Metallurgy and Engineering Alloys”.

Dr. Zinovieva's  explanation for constitutional super cooling is superb.  But it comes to the thermal supercooling she didn't mentioned very much. The solidification is not a simple bulk  phase transformation. In ordinary sense it involves nucleation state where the creation of surface or interface between the embryo and the liquid environment  involves very high activation energy barrier unless you have heterogeneous nucleation, otherwise  (homogeneous) it requires large localized temperature fluctuations to be over come even though bulk Gibbs free energy of transformation becomes negative due to the fact that  instantaneous temperature may be few degrees below the equilibrium solidification temperature. Here, only one thing can be done is  to lower the temperature of liquid as such that Gibbs free energy of transformation should be large enough negative to compensate the activation energy for solidification.  The temperature difference between  the actual solidification starts  and the one obtained from the  equilibrium phase diagram   is called  supercooling temperature, which is negative quantity.  This was first time measured by Turnbull of Harward , and he found del T= about  100C  'small droplets of Hg, copper, silver, gold and many other metals  etc.

''At the 1948 dinner meeting of the Science of Metals Club, Turnbull told his audience that he had undercooled liquid mercury several tens of degrees below its thermodynamic crystallization temperature by dispersing it into small droplets.'^'.... 

The book "Phase Transformations in Metals and Alloys" by Porter and Easterling gives a a good explanation of what happens at the solid liquid interface during solidification.

I agree with all of the above responses. I would like to refer you to the reference " S.A. David and J.M Vitek "Correlation between solidification parameters and weld microstructure " International Materials Review, 1989, Vol 34 no 5, p213.The paper describes undercooling and nucleation barriers etc

'Fundamentals of Solidification, 3rd edition' by Kurz and Fisher gives a very good explanation of thermal and constitutional undercooling.The undercooling is the difference between the melt temperature and either the thermodynamic melting point (for a pure substance) or the liquidus (for an alloy). For solidification (solidification = nucleation +growth) to occur undercooling must arise to create the thermodynamic driving force for phase change.

In the particular case of a moving solid-liquid interface in a pure substance, 2 different situations can arise: either there is a positive temperature gradient ahead of the growth interface (i.e. directional solidification, Bridgman, etc...) or a negative temperature gradient. For a positive temperature gradient the latent heat of fusion flows through to the solid only and growth is always stable. In case there is a negative temperature gradient in front of the growth interface, the difference between the interface temperature and the temperature far away in the melt is called thermal undercooling. In this case (e.g. growth in an droplet undercooled because nucleation was avoided) the interface is always instable and dendritic growth occurs.

For an alloy the constitutional undercooling arise because a negative concentration gradient arises ahead of the growth interface due to pile up of the solute rejected in the melt (in the usual case that the solute is more soluble in the melt than in the solid). This negative concentration gradient may be seen as a positive 'melting point gradient' for the liquid ahead of the growth interface (since melting point and concentration are related via the liquidus line in the phase diagram). Thus the interface is at higher solute concentration and has a lower melting point than the melt far from the interface. This melting point change with respect to the melting point of the composition far from the interface is called constitutional undercooling. In this case, even under positive temperature gradient (Bridgman, directional solidification etc...) growth can be unstable (i.e. dendrite growth occurs) if the gradient of melting point change is higher than the temperature gradient. (see http://www.doitpoms.ac.uk/tlplib/solidification_alloys/undercooling.php for a useful scheme)

Thermal  supercooling occured  during solidification of pure liquid metal . Here heat is removed from liquid metal  by convection process or conduction process from solid metal & mould - liquid surface through container wall.

Constitutional Supercooling happened during soldification of alloy liquid metal. It is a  negative temperature gradient ahead of the interface between liquid and solid boundary surface. Here solid - liquid interface is unstable and solute is rejected from solid surface to near boundary liquid .As a  result temperature difference occured.

Everybody has tried to explain what constitutional supercooling is. Can anybody give a definition in one or two lines max.  

Constitutional super cooling is the difference between the actual temperature and the liquidus lins.

Prof. Doru M. Stefanescu explains the thermal undercooling in his book as follow:

"As long as nucleation of solid and subsequent growth of these nuclei is rather fast, the only S/L interface undercoolings for the pure metal are kinetic and curvature. However, if nucleation difficulties are encountered, or if growth of the solid lags heat transport out of the liquid, an additional undercooling, thermal undercooling occurs."

For example, if you increase the cooling rate, it increases the probability of thermal undercooling occurrence.