A system with many microscopic components (for example, a gas, a liquid, a solid with many molecules) that is isolated from all forms of energy exchange and left alone for a ``long time'' moves toward a state of thermal equilibrium. A system in thermal equilibrium is characterized by a set of macroscopic quantities that depend on the system in question and characterize its ``state'' (such as pressure, volume, density) that do not change in time.
Two systems are said to be in (mutual) thermal equilibrium if, when they are placed in ``thermal contact'' (basically, contact that permits the exchange of energy between them), their state variables do not change.
If system A is in thermal equilibrium with system C, and system B is in thermal equilibrium with system C, then system A is in thermal equilibrium with system B.
The point of the Zeroth Law is that it is the basis of the thermometer. A thermometer is a portable device whose thermal state is related linearly to some simple property, for example its density or pressure. Once a suitable temperature scale is defined for the device, one can use it to measure the temperature of a variety of disparate systems in thermal equilibrium. Temperature thus characterizes thermal equilibrium.