a second-order phase transition
Not every transition is abrupt. Heat a magnet through its Curie temperature and its magnetization fades smoothly to zero with no latent heat, no coexisting phases, no sudden jump — yet something dramatic still happens: the heat capacity spikes and the magnetic susceptibility diverges. Cooling helium into a superfluid, or a metal into a superconductor, behaves the same way. These are continuous, or second-order, phase transitions.
In the original Ehrenfest scheme a transition is second-order when the Gibbs free energy and its first derivatives (entropy, volume) are continuous, but its second derivatives — heat capacity C_p, compressibility, susceptibility — are discontinuous. The modern picture is richer: a continuous transition is characterized by an order parameter that rises continuously from zero (e.g. the magnetization), a diverging correlation length, and power-law critical exponents, described by Landau theory and the renormalization group and grouped into universality classes.
Because there is no latent heat and no interface to nucleate, continuous transitions show no hysteresis and no phase coexistence. Honest caveat: Ehrenfest's tidy 'the nth derivative jumps' hierarchy is largely obsolete, because at most real continuous transitions the second derivatives do not merely jump but diverge — which is why the term 'continuous transition' is now preferred to 'second-order'.
In a ferromagnet the magnetization M is the order parameter: below the Curie temperature T_c it is nonzero, and as T rises to T_c it falls continuously to zero (as M ~ (T_c - T)^beta near the transition). There is no latent heat, but the magnetic susceptibility diverges at T_c — the fingerprints of a continuous transition.
A continuously vanishing order parameter and a diverging response function mark a second-order transition.
Calling these 'second-order' after Ehrenfest is a historical label; at genuine critical points quantities like the heat capacity typically diverge rather than showing a finite discontinuity, so 'continuous transition' is the more accurate modern name.