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The Big Bang is a theoretical event when the expansion of the universe began from a single point, i.e., began assuming spatial dimensions. Observation shows the observable universe expanding, and if that expansion has "always" happened, projecting back sufficient time, there would be an instant when the universe expanded from a single point. Current observation places the moment at about 13.8 billion years ago, a precise determination requiring knowledge of the expansion rate over the course of this expansion, i.e., knowing the behavior of the Hubble parameter over time.
Though the finite speed of EMR allows us to observe past times by looking long distances, our ability to observe detail all the way to this theoretical Big Bang is limited, limiting what we can say about the very beginning. However, the notion that expansion has persisted since the general time indicated by observed expansion (the Big Bang theory) is well-established by observation and generally accepted, and no alternative scenario to a Big-Bang instant has any more evidence. Cosmology does explore scenarios all the way back to the apparent instant of the Big Bang, applying the established (and seemingly-likely) laws of physics. Some observed characteristics of the universe today are consistent with events that experimentally-established physical laws suggest would happen in that era, all the way back to a small fraction of a second after such a Big Bang.
Before the Big Bang was postulated, the tacit assumption was that the universe remains the same, and when a notion of expansion was seriously introduced (which was based upon observation and was something of a shock), the older assumption came to be termed a steady-state universe. It was this older assumption that had led Einstein to include a feature in his general relativity (GR) theory to keep the universe the same size (i.e., the cosmological constant in GR's field equation). Then when the universe's apparent expansion become evident, Einstein set aside the feature. However, for some time, serious theories of how the universe might be "steady state" continued to be developed. Eventually, features of the universe were observed that could be explained by the Big Bang theory but not by competing theories.
The term standard Big Bang (SBB) refers to a "working model" version of the Big Bang theory specifically assuming that physics and physical constants at the time were the same as now. (Other theories might speculate that some constants have varied according to the size/spread of the universe.)
Note there have been further theoretical developments beyond the presumption of expansion from a Big Bang, and now the term Big Bang theory often means the current such theory, which includes newer developments that are well-accepted, such as inflation, dark matter, dark energy, etc.
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