Similarity between Ecell and equilibrium constant Kc
$E^\circ_{cell}$ and $K_c$ are related. Both indicate which direction of the reaction is spontaneous. Both tell you something about the position of equilibrium.

However, $E^\circ_{cell}$ is a directly measurable value. If you construct an electrochemical cell as using a zinc electrode immersed in a 1.0 M solution of say zinc chloride and a copper electrode immersed in a 1.0 M solution of copper (II) chloride with a salt bride of sodium chloride connecting the solutions, a voltmeter connected between the two electrodes will read the value of $E^\circ_{cell}$, at least initially. The voltage will change as the reaction progresses, but the initial voltageis $E^\circ_{cell}$ which is defined for standard conditions of 1.0 M concentration, 1 bar of pressure, and 25 $^\circ$C.

$$\ce{Zn|ZnCl2}(1.0 \text{ M})\parallel\ce{CuCl2}(1.0 \text{ M})|\ce{Cu} \ \ \ \ E^\circ_{cell}=1.1\text{ V}$$

The relationship between $E^\circ_{cell}$ and $K_c$ is the Nernst Equation. Note the use of both $E^\circ_{cell}$, the standard cell potential and $E_{cell}$, the observed cell potential under other conditions. The remainder of the alphabet soup is defined below.
$$E_{cell}=E^\circ_{cell}-\frac{RT}{zF}\ln{Q}$$


$E^\circ_{cell}$ - Standard cell potential
$E_{cell}$ - nonstandard cell potential at other conditions
$R$ - the ideal gas constant (which appears here because it is equal to Avogadro's number times Boltzmann's constant
$T$ - the temperature in kelvins
$z$ - the number of moles of electrons being transferred in this redox process. If we wanted to use the number of individual electrons, then we would need to replace $R$ with $k_B$
$F$ - Faraday's constant
$Q$ - the reaction quotient


We need to use $Q$ in the Nernst equation, because this equation can relate the cell potential to any range of conditions, not just equilibrium. 


  At equilibrium $Q = K_c$ and $E_{cell} = 0 \text{ V}$ (there is no net change in the reaction composition, so there is no potential difference across the half-cells.


Thus, the Nernst Equation becomes:

$$0=E^\circ_{cell}-\frac{RT}{zF}\ln{K_c}$$
$$E^\circ_{cell} = \frac{RT}{zF}\ln{K_c}$$


  To determine the values of $E^\circ_{cell}$ and $K_c$ for the reverse reaction...


For $E^\circ_{cell}$:

$$E^\circ_{cell} = -E^\circ_{rev}$$

For $K_c$

$$K_{rev} = \frac{1}{K_c}$$