Definitional Issues
A theory about the beginning of the Universe and similar matters is generally called a "cosmology" and a fairly complete, but not comprehensive, list can be found
here.
Exactly how many alternatives there are to the Big Bang is more of a definitional issue than anything. Where a cosmology model, like the paradigm of the LambdaCDM model, and a "cosmology" itself begin and end is ill defined. A LambdaWDM model (with warm dark matter rather than cold dark matter which is identical except for the dark matter particle mass which is smaller in warm dark matter theories) might be viewed as a variant of the same cosmology or same class of cosmologies as a LambdaCDM model, for example.
Lumpers would call almost all observationally viable or not scientifically ruled out cosmologies variations on the Big Bang theory or versions of it. Splitters would argue that each variation is a different theory or class of theories.
Cosmologies To Ignore
Many cosmologies are patently unscientific and rooted in religion or mythology. Many other cosmologies are no longer accepted by the scientific community, even though they were once considered credible scientific theories until new information ruled them out, such as the steady state theory and tired light theories. Both have only historical interest from an astrophysicist's perspective.
Viable Cosmologies Compared
The differences between the
cosmologies that remain viable in light of current observational and experimental evidence primarily differ from each other (1) with respect to cosmological inflation, which is an extremely rapid expansion of the Universe sometime in a first very tiny fraction of a second (of which there are hundreds of variations), and (2) with respect to what (if anything) the cosmology says about possible points of time prior to a "Big Bang"-like event, for example, in the case of bouncing/cyclic cosmologies, or in the case of two sided cosmologies (with a parallel universe in which time runs in the opposite direction from that of our own Universe and is matter dominated before the Big Bang, see, e.g.
here).
There are also (3) various kinds of many-worlds cosmologies and (4) cosmologies in which other stuff outside the observable Universe (defined rather than mechanically to exclude various indirect hints of its existence) is important.
Likewise, cosmologies differ meaningfully in their dark sector (i.e. dark matter and dark energy) content, and in their operative laws of gravity to the extent that they derived from subtle modifications of general relativity that can't be ruled out experimentally. But the vast majority of these variations are still squarely within the overall larger Big Bang hypothesis. Whether a Big Bang-like cosmology model like LambadaCDM and a modified gravity cosmology without dark matter or dark energy count as subtypes of one Big Bang cosmology, or as two different cosmologies, is largely a matter of opinion.
One could also arguably call various string theory approaches (such as higher dimensional brane worlds in which our existence is confined to an observable four dimensional universe) a form of a cosmology, although again, this not really contrary to the Big Bang Theory so much as it puts the Big Bang in a large multidimensional context.
Why Consider Alternatives?
Generically, alternatives to the plain vanilla "Big Bang" with all things existing originating then, are devised to be subtly different from the Big Bang: (1) in order to solve issues that might otherwise be explained with
cosmological inflation, (2) by addressing issues related to matter creation (e.g.
the matter-antimatter asymmetry of the universe), (3) by explaining anisotropy (i.e. asymmetry in the distribution of mass and energy) or violations of the "
cosmological principle" (i.e. that there are no observable irregularities in the large scale structure of the universe at a large enough scale) that astronomy observations may hint at, (4) by providing some sort of probabilistic motivation for the values of physical constants or (5) for some sort of explanation of the "weirdness" of quantum mechanics.
