What is Molar Conductivity of an electrolytic solution
I will answer your question about the definition of molar conductivity and why it increases with decreasing concentration.

As Jerry pointed out, you can probably learn what you need about equivalent conductivity using a Google search.

The molar conductivity of an electrolyte solution is the conductivity of the solution divided by the molar concentration of the dissolved electrolyte. 

Conductivity ($\kappa$, in $\text{S/cm}$) is proportional to the square root of the concentration of ions ($\sqrt{[\text{ions}]}$, withy $\ce{[ions]}$ in $\text{mol/L}$). In the following equation, A is an empirical proportionality constant with units needed to convert from $\text{M}^{1/2}$ to $\text{S/cm}$, namely $\frac{\text{S/cm}}{\text{M}^{1/2}}$

$$\kappa = A\sqrt{\ce{[ions]}}$$

The ion concentration is related to the electrolyte concentration ($C$) by the van't Hoff factor ($i$).

$$[\ce{ions}]=iC$$

$$\therefore \ \kappa = A\sqrt{iC}$$

The molar conductivity $\kappa_M$ is the conductivity divided by the molar concentration of electrolyte ($C$).

$$\kappa_M=\frac{\kappa}{C}=\frac{A\sqrt{iC}}{C}=A\frac{\sqrt{i}}{\sqrt{C}}$$

As $C$ decreases, so does $\sqrt{C}$, and thus $\frac{1}{\sqrt{C}}$ increases, while $A$ remains constant. The van't Hoff factor changes a little with concentration, but not dramatically. ($A$ probably also varies a little with concentration. It certainly varies with temperature.)

For example, if you have a series of solutions of sodium chloride, concentration decreases faster than conductivity, so conductivity divided by concentration increases. Note that the conductivity data are fabricated with an arbitrary value of $A$. I could not find the specific value of $A$ for sodium chloride. The van't Hoff Factors come from the "Colligative Properties" Chapter of Chemistry: Atoms First by Burdge and Overby, McGraw Hill, 2012.