Is sea water more conductive than pure water because "electrical current is transported by the ions in solution"? Apparently, electrical charge is transported by the ions dissolved in water, is this true?
 A: Yep. Pure water is an extremely bad conductor of electricity, it has very few ions. Water with an electrolyte (like NaCl) is a much better conductor of electricity; as the ions can migrate. Migration of ions is just like migration of electrons. If you place an imaginary surface inside the cell, there will be net negative charge crossing over to the positive terminal and vice versa. This is just like a current. Since there is net current inside, its conducting.
The equivalent conductance (a loco* chemistry concept) of a solution is simply the sum of the conductances of its constituent parts (Kohlrausch law). Here, $\Lambda$ denotes equivalent conductance of a portion of the solution, and $\lambda$ is the same for ions. Just a notation.
For pure water,
$$\Lambda_{H_2O}=\lambda_{H^+}+\lambda_{OH^-}$$
Now, since the concentrations of $H^+$/$OH^-$ are small ($10^{-7} M$ at STP), the $\lambda$s and thus $\Lambda_{H_2O}$ are pretty tiny.
For water with salt in it, we get 
$$\Lambda_{soln}=\Lambda_{H_2O}+\Lambda_{NaCl}=\lambda_{H^+}+\lambda_{OH^-}+\lambda_{Na^+}+\lambda_{Cl^-}$$
Since $NaCl$ nearly dissociates completely, we get large $\lambda$s, and thus $\Lambda_{soln}$, which can be related to conductivity (in the aforementioned loco way).
So, pure/distilled water is an extremely bad conductor, while impure water with ions in it is a good conductor
 * Loco because they assume a 1 m cell throughout, and don't keep the necessary $\text{m}^{-1}$ or whatever in their units. Due to this fixing of parameters, yes, we get that $\text{Area of plates}=\text{volume}$, which lets us relate it to concentration; but this gives us predictions for a specific case; when length of the cell is 1 m only. For some reasons these predictions are blindly applied to the general case. The whole thing gets confusing if you try to visualise it.
