Any massive cosmic object that experiences some kind of acceleration will generate gravitational waves.
MORE: Einstein's General Relativity Still Put to Test
Black holes are the most massive and dense objects known to exist in the universe and are hotbeds of gravitational wave activity, especially if they collide and merge. Merging black holes are thought to be the key growth mechanism behind these gravitational behemoths -- when two galaxies merge, their central supermassive black holes will begin orbiting one another, eventually spiraling in and then colliding to form an even bigger black hole. In this scenario, gravitational waves will be emitted from the spiraling black holes long before they collide, but as the objects draw closer, gravitational wave energy will increase, sapping more and more orbital energy from the pair until they collide, ringing like a "bell" after they merge.
Another energetic phenomenon that would generate a rapid eruption of gravitational waves are supernovas. After a massive star runs out of hydrogen fuel it implodes, succumbing to massive gravitational pressure. The resulting explosion will fire a pulse of gravitational waves that will wash through spacetime.
Gravitational waves will also be generated by rapidly spinning objects, but there's a catch. Only massive spinning objects that are asymmetric (i.e. not symmetrical) will produce gravitational waves in a periodic pattern. For example, a rapidly spinning neutron star with a clump of material bulging from one hemisphere will "stir up" spacetime to generate gravitational waves. A perfectly symmetrical neutron star, however, will not generate gravitational waves. The easiest way to visualize this is to imagine spinning an oval-shaped football on the surface of a swimming pool; as the football spins, it will create ripples across the water. A spherical spinning soccer ball, on the other hand, will create very few ripples on the surface.
MORE: BICEP2 Gravitational Wave 'Discovery' Deflates
The Big Bang is also theorized to have generated a powerful hum of gravitational waves when the universe began, nearly 14 billion years ago. However, these primordial gravitational waves are unlikely to be directly detected as their signal is too weak in the modern universe. But efforts are under way to detect their presence in the "background glow" of the Big Bang. Projects like the BICEP2 telescope at the South Pole are looking for a very specific type of polarization in the cosmic microwave background (CMB) that is thought to be caused by primordial gravitational waves. Despite recent announcements to the contrary, this signal has yet to be detected.