Very briefly, we have three types of equilibrium:
Thermal equilibrium means equality of temperatures. So no heat transfer.
Mechanical equilibrium means equality of forces. So no mechanical movement.
Chemical equilibrium means equality of chemical potentials. So no changes in the concentration of particles of any specific type, either because of (1) transfer from one subsystem to another, or (2) their creation or destruction in chemical reactions.
The term kinetic equilibrium is not a fourth item on this list, it refers to something that is related to thermal equilibrium. I found a good discussion here. Here is an excerpt defining the term, though it's definitely worthwhile to read the rest of the discussion there:
Kinetic equilibrium obtains when reactions that exchange energy between particles (such as collisions) occur rapidly compared to the time-scale under which conditions are changing.
ADDENDUM: a good clarifying question was asked in the comments, which I think will make this answer more complete.
Q. Can you maybe expand a bit on the first subtype of chemical equilibrium? I am unsure what a "subsystem" could refer to in the early universe.
A. The division of a system into subsystems is essentially arbitrary. You can view two subsystems, which may or may not be in any of those types of equilibrium with each other, simply as two neighboring regions of space. So if, for example, there is no net transfer of electrons between those two regions, then we would say these two regions are in chemical equilibrium of the first type with respect to electrons. Note: saying "of the first type" is not standard terminology. Usually people simply talk about systems having equal chemical potentials for electrons (or any other particle or molecule).
