To take a quote from Wikipedia,
The Big Bang is the cosmological model of the universe, originally conceived to describe its origins, that is best supported by all cosmological observations. The essential idea is that the universe has expanded from a primordial hot and dense initial condition at some finite time in the past and continues to expand to this day. Observational evidence for the expansion of the universe came first from Edwin Hubble's 1929 discovery that galactic distances are proportional to their redshifts.[1] Since the universe is seen to be expanding today, theoretical cosmologists can make detailed predictions of the universe's denser and hotter past. A major feature of Big Bang cosmology is its prediction of the conditions of an early universe having extreme densities and temperatures that today can only be probed in high-energy environments. The results of Big Bang predictions have been found to conform accurately and precisely to a variety of astronomical observations.
In fiction, a Big Bang is usually portrayed as a massive release of energy, instead of an expansion of spacetime itself. A Big Bang is generally considered to be a universe level attack (however this should be treated on a case by case basis).

Characters that can release Big Bangs or Big Bang level energy:

- The Anti - Spiral, from TTGL (somewhat disputed)
- Akira and Tetsuo, from Akira (disputed due to size and vague details regarding said Big Bang)
- Apocalymon (Digimon)
- Bernkastel and Lambdadelta from Umineko no Naku Koro ni
- The Destroyer from Vampire Hunter D (disputed)
- The Downstreamers from the Manifold series
- Celestials, from Marvel Comics
- Chronos, from Saint Seiya
- Cube Beings, from Marvel comics
- Franklin Richards, from Marvel Comics
- Galactus, from Marvel Comics
- Getter Emperor, from Getter Robo
- Imperiex, from DC Comics
- Ideon (Space Runaway Ideon)
- The Monads, from Xeelee Sequence
- Three Gold Saints from Saint Seiya using Athena Exclamation (note it is a Big Bang on a smaller scale)
- Monarch, from DC Comics
- Firestorm, from DC Comics (only as Elemental Firestorm)

A scientific model. Needs more fancy explosions
A scientific model. Needs more fancy explosions

See also:

Big Crunch
Big Rip