A black hole is like a spinning drop of viscous liquid. Normally, such a drop tends to stay whole: splitting would reduce its surface area, which contradicts the law of non-decrease of entropy (a measure of disorder, like a broken cup doesn't glue itself back together). For non-rotating black holes, discovered by Karl Schwarzschild, splitting is impossible. However, with furious rotation, centrifugal forces stretch the black hole, and a microscopic fragment can pinch off. This process resembles nuclear fission in the liquid-drop model. Physicists have also considered worlds with extra dimensions, where spacetime curvature behaves differently, and black holes can become unstable, like long liquid threads ready to snap into pieces. Such decay could alter the population of primordial black holes, born in the young Universe.
Merging black holes, on the contrary, always increases the total area: this process resembles the merging of mercury droplets and is accompanied by powerful gravitational waves.