Chandra X-ray observations revealed the presence of cold fronts -- sharp contact discontinuities between gas regions with different temperatures and densities -- in the centers of many, if not most, relaxed clusters with cool cores. We use high-resolution simulations of idealized cluster mergers to address the origin of such fronts. Typical observed gas density and temperature profiles are used as initial conditions. We find that these cold fronts are due to sloshing of the cool gas in the central gravitational potential well, which is easily set off by any minor merger and can persist for gigayears. The only necessary condition for their formation is a steep entropy drop in the gas peak, as found in the cooling flow clusters. Most interestingly, the fronts form even if the infalling subcluster has no gas during core passage. Gravitational disturbance from a pure dark-matter subcluster sets the main mass peak (gas and dark matter together) in motion relative to the surrounding cluster gas. A rapid change in the direction of this motion during the core passage causes a change in ram pressure, which pushes the cool gas temporarily away from the dark matter peak. The cool gas then falls back and starts sloshing in the potential minimum, generating mushroom-like cool edges at each oscillation. The edges slowly move outwards even as the densest gas turns around. If the subhalo had a nonzero impact parameter, the cool gas acquires angular momentum while offset from the center, and does not fall back radially. The resulting cold fronts combine into a characteristic spiral pattern, which initially does not represent any coherent spiraling motion, but evolves into a spiral inflow. There is little visible disturbance outside the cool core in such a merger. The picture is qualitatively different if the subcluster contains gas during core passage. Then the dominant agents are the shock front and stripped gas from the subcluster, which decouple most or all of the cool central gas of the main cluster from the dark matter peak via a "ram-pressure slingshot". Subsequently, some of that gas, and often the cool gas from the subcluster, falls back to the center and starts sloshing in the potential minimum. Such a merger creates global disturbance in the ICM readily visible in the X-ray image long after the first core passage. We conclude that cold fronts at the centers of such exceptionally relaxed clusters as A2029 or A1795, often spiral or concentric-arc in shape, are probably caused by encounters with small subhalos stripped of all their gas at the early infall stages.