Cosmic time is a time scale that assumes the origin of the 4 dimensional space time at the big bang. Because Big Bang is a sudden expansion of the universe, in this scale, the universe is assumed to be homogeneous.
As you may know, since Einstein we know that Time is not absolute, but it may flow differently for different entities, usually called “observers” although that name does not imply the existence of any rational beings around. Time simply flows at different relative rates for entities which have relative motion or which reside in gravitational wells of different magnitude.
So in order to assess things like the age of the universe or other cosmological arguments we need to overcome that subjective aspect of Time.
Cosmic Time is defined as the rate of time experienced by clocks which move along the Hubble flow, that is, the expansion of space in our universe. This definition explicitly excludes any effects on the actual ticking of physical clocks due to their motion relative to other clocks or due to their being under the influence of significant gravitational fields.
So it can be considered as a truly objective, non-observer-dependent, rate of the flowing of Time, and it is used for astronomical or cosmological arguments such as defining the age of the universe.