The property of quarks due to which they never exist as free particles but always exists in bound colour-less states.
This property is an empirical fact which is believed to follow from the fundumental theory of quantum chromodynamics (QCD) which is supposed to account for quarks and the strong nuclear force. There is, however, no theoretical derivation of this property to date, at least not a one without further assumptions. A possible suggested mechanism for quark confinement is the so called dual Meissner effect. The idea is that due to condensation of chromo-magnetic monopoles in the QCD vacuum, it acts as a dual superconductor. Therefore, in analogy with the usual superconductor which is known to confine magnetic monopoles (if those exist), the QCD vacuum confines
chromo-electric charges, meaning anything that has colour. This can be thought of as follows: the QCD vacuum doesn't admit the chromo-electric field into itself, and as a result this field forms thin flux tubes as opposed to the usual electric field which spreads around space giving rise to the Coulomb law. The binding energy between two coloured objects then becomes directly proportional to the length of the flux tube, i.e., the distance between them. This means they are in effect in an infinitely deep potential well, as opposed to the electric case where there is a finite escape velocity (notice the analogy with gravitation and Newton's law). In fact, if you pull two bounded quarks (a quark and an anti-quark, more precisely) far enough from each other, the energy you invested will lead to the formation of a new quark-anti-quark pair and the meson will become two mesons. Imagine the flux tube connecting the quark to the anti-quark splitting in the middle and the new quark and anti-quark appearing on the newly formed endpoints.
Some attempts have been made to derive quark confinement from QCD using numerical methods (lattice gauge theory), with a moderate amount of success.
Theorists have been claiming for some time that at a high enough temperature (much higher than the temperature at the heart of a hydrogen bomb explosion) a phase transition occurs into a deconfined phase. In this phase there is no longer quark confinement, the quarks exist in free form, and the strong nuclear force is a lot more similar to the electric force. Some evidence for this phase transition have already been obtained at CERN. The new phase has been named quark-gluon plasma.