Magnetic fields are transported and tangled by turbulence, even as they lose identity due to nonideal or resistive effects. In a balanced field, lines undergo stretch-twist-fold processes. The curvature field, a scalar that measures the tangling of the magnetic field lines, is studied in detail here, in the context of magnetohydrodynamic turbulence. A central finding is that the magnitudes of curvature and magnetic field are anticorrelated. A high curvature colocates with a low magnetic field, which gives rise to power-law tails of the probability density function of the curvature field. The curvature drift term that converts magnetic energy into flow and thermal energy largely depends on the curvature field behavior, a relationship that helps to explain particle acceleration due to the curvature drift. This adds as well to evidence that turbulent effects most likely play an essential role in particle energization since turbulence drives stronger tangled field configurations, and therefore curvature.