For ground-based detectors like Advanced LIGO, there are 4 main astrophysical/cosmological sources that could produce GWs. These are coalescing compact binaries, rapidly spinning neutron stars, asymmetric core collapse supernovae, and the Big Bang. In addition to these four, we also actively search for GWs associated with gamma-ray bursts, fast radio bursts, magnetar flares, soft gamma repeaters, cosmic string cusps, etc. And of course, there might be unexpected surprises!
This type includes stellar systems of binary black holes (BBHs), binary neutron stars (BNSes), and neutron star - black holes (NSBHs). For LIGO, these systems produce effectively transient signals when the two bodies in a system plunge into each other and produce GW signals that fall into LIGO's sensitive frequency range. For BBHs, signals last for usually a fraction of its last second before merger. For BNSes, signals last for tens to hundreds of seconds leading up to merger. For NSBHs, signals last for anything in between. These show chirp-like signals in time-frequency plots in LIGO, which increase in both frequency and amplitude as they draw closer to merger.
CBCs have been the only detected astrophysical source up till LIGO Observing Run 3. Stellar black holes are not observable in EM waves. GWs, instead, are great messengers for their existence and a great tool to study their properties and dynamics. On the other hand, neutron stars are still very much an enigma. Just like how the first LIGO BNS signal, GW170817, enabled us to learn so much about neutron stars, GWs will continue guiding us to solve the mystery of neutron stars.
Pulsars, or spinning neutron stars, are so very dense. (In fact, their high densities are one reason we don't know much about them, as we cannot reproduce such dense environments in labs on Earth.) Added with the fact that they are rapidly spinning, pulsars are mostly spherically symmetric. Perfectly symmetric pulsars don't emit GWs. But in the case where there exist tiny deformations on a pulsar (a few centimeters of a bump is all it takes!), it will no longer be spinning spherically symmetrically. Pulsars will then emit GWs that we describe as continuous waves. Continuous waves are of fairly constant amplitudes and frequencies and are comparably weaker than other types.
Unless there is a complete spherical symmetry in the explosion of a supernova, it will generate GWs. Such GWs are not well modeled, as supernovae are not well understood. Asymmetric core collapse supernovae could leave burst-like signals in the LIGO frequency band.
Many cosmology models suggest that the Universe went through an inflation epoch not long after the Big Bang. Inflation is a period of exponential expansion of the early Universe. If this expansion was not symmetric homogeneously, a cosmic Gravitational Wave Background (GWB) was likely created. Just like detecting the EM relic radiation from the early Universe, the Cosmic Microwave Background (CMB), detecting a cosmic GWB can help us learn about the origin and the early history of the Universe. GWs cannot be absorbed, scattered, or shielded. The cosmic GWB could potentially take us back to ~10-36s after the Big Bang, compared to 380,000 years after the Big Bang as learned from the CMB. Just like the CMB, the cosmic GWB is expected to be homogeneous and isotropic, which constitute stochastic GWs.
With GWs observed using prescribed gravitational waveforms computed through numerical relativity, once again, we have proved that "Einstein was right". BUT only for the moment being!
We are testing properties of detected GWs and looking for deviations from the predictions of GR, including the speed of GWs, Lorentz and parity violations, and more polarization states than just the tensor polarization.
GWs provide unique highly-dynamical and strong-field tests of GR. We are searching for deviations of detected signals from numerical relativity calculations, which could point us to alternative theories of gravity.
From detected GWs, we can extract physical parameters of exotic stellar objects and cataclysmic events such as neutron stars, black holes and supernovae. For compact binaries, masses, spins, and eccentricities can inform us about the binary formation channels. For neutron stars, tidal deformation and other parameters like spins encode the key to dense nuclear equation of state. For supernovae, GW observations contribute directly to understanding the core-collapse supernova mechanism.
We can also learn about the underlying populations of sources from many detected individual events. For example, binary masses, spins, eccentricities, orientations, etc. We can also infer the astrophysical rates of these events.
We can make cosmological measurements, such as the Hubble constant, using detected GWs.
A GW source is a "standard siren" of known loudness, meaning that we can obtain a measurement of the luminosity distance to a source. If combined with a redshift measurement, either from an observation of an EM counterpart, or from statistical calculations by overlaying galaxy catalogs onto the source localization, we can probe the expansion history of the Universe.