There is another closely related post here , and I've also read the referenced wiki article including applications of chelation, but I still don't see what's so special about chelation. I understand the structure of the bond - the 'capturing' of a metal cation by an organic agent, but I still fail to see why this chemical reaction might be so special relative to other possible organic chemical reactions with a metal cation.
Why is chelation so special?
Chelation is an entropy driven reaction, i.e. complexes with polydentate ligands are more stable.
Consider what happens when you dissolve copper sulfate in water: \begin{align} \ce{CuSO4 + 6H2O <=> [Cu(H2O)]^2+ + SO4^2- (aq)}\\ \end{align}
This complex forms and with it comes some kind of ordering of the water molecules around it. Dissolution is often favoured because of the solvation energy. Now consider a bidentate ligand like ethylene diamine, $\ce{H2N-(CH2)2-NH2}$, short $\ce{en}$. Adding this ligand to a copper sulfate solution will result in a ligand exchange: \begin{align} \ce{[Cu(H2O)]^2+ + 3en <=> [Cu(en)3]^2+ + 6H2O}\\ \end{align}
In this reaction you increase the number of particles from four on the left side to seven on the right side. Taking into account that you also disturb the order of the surround water molecules, this reaction is very much favoured by entropy. In an attempt of zeroth order approximation, one could assume that entropy roughly doubles. Since the Gibbs energy is inversely correlated to the entropy, this means, that the reaction will be spontaneous. This is usually referred to as the chelate effect.$%edit$
To complement Martin's answer with regard to the applications: chelation is particularly useful in the body with regard to iron transportation in siderophores and porphyrin rings and also in medicine where excess metal ions build up such as Wilkins disease ($\ce{Cu^2+}$) etc. Typically you see drugs with connected phenoxides, hydroxamates or EDTA fragments exactly for this purpose directed to iron deposits in the liver. Upon complexation you remove the chelated product easily. Without the Gibbs drive, metal plaques up. In addition catalysis and organometallic chemistry is full of chelation chemistry.
