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Cosmic time (also known as time since the big bang) is the time coordinate commonly used in the Big Bang models of physical cosmology. It is defined for homogeneous, expanding universes as follows: Choose a time coordinate so that the universe has the same density everywhere at each moment in time (the fact that this is possible means that the universe is, by definition, homogeneous). Measure the passage of time using clocks moving with the Hubble flow. Choose the big bang singularity as the origin of the time coordinate.
Cosmic time   is a measure of time by a physical clock with zero peculiar velocity in the absence of matter over-/under-densities (to prevent time dilation due to relativistic effects or confusions caused by expansion of the universe). Unlike other measures of time such as temperature, redshift, particle horizon, or Hubble horizon, the cosmic time (similar and complementary to the comoving coordinates) is blind to the expansion of the universe.
There are two main ways for establishing a reference point for the cosmic time. The most trivial way is to take the present time as the cosmic reference point (sometimes referred to as the lookback time) or alternatively, take the Big Bang as (also referred to as age of the universe). The big bang doesn't necessarily have to correspond to a physical event but rather it refers to the point at which the scale factor would vanish for a standard cosmological model such as ΛCDM. For instance, in the case of inflation, i.e. a non-standard cosmology, the hypothetical moment of big bang is still determined using the benchmark cosmological models which may coincide with the end of the inflationary epoch. For inflationary models, it is not possible to establish a well defined origin of time before the big bang since the universe does not require a beginning event in such models. For technical purposes, concepts such as the average temperature of the universe (in units of eV) or the particle horizon are used when the early universe is the objective of a study since understanding the interaction among particles is more relevant than their time coordinate or age.