$K$, the standard thermodynamic equilibrium constant, is computed from $\Delta G^\circ$, using
$$\Delta G^\circ = -RT\log K \tag1$$
Generally speaking, $K$ in equation (1) is unitless. Its value depends on the specified reference standard states and $T$ (and obviously on the equilibrium activities of reactants and products). Both $K$ and $\Delta G^\circ$ change (and are certainly allowed to) if you change the choice of standard states for any species.
You can also compute an equilibrium constant from a value of $\Delta G^\circ$. To do this reverse operation you need to know the standard state ($p^\circ=\pu{1 bar}$, $c^\circ=\pu{1 M}$, or $m^\circ=\pu{1 molal}$, $\chi^\circ=\pu{1}$) associated with each species in the equilibrium equation. At equilibrium each product/reactant is at the same temperature and in the same phase as in its associated reference standard state (if not at the same partial pressure, concentration, or mole fraction).
When you use either $K$ or $\Delta G^\circ$ in practice, you need to know the standard state of each species (this is often apparent from additional information that is provided with these parameters). If you are working with a gas-phase reaction then you can refer to $K$ as $K_p$. If you are using standard states then all reference states are $\pu{1 bar}$ $^\dagger$, and partial pressures computed directly from $K_p$ are in $\pu{bar}$ units (these can obviously be converted later).
To compute $K_c$ from say $K_p$ you have to convert units with an appropriate conversion factor. This is described in the linked post. Same applies to problems where $K_\chi$ is provided, where concentrations are described in mole fraction units. Note that if you change units, for instance to compute $K_c$, then you have (perhaps implicitly) changed the reference state (perhaps to a nonstandard state), and in that case $\Delta G^\circ$ also changes. This is also explained in an answer to the linked post.
$\dagger$ Components whose activity is constant can usually be ignored
