With a strong electrolyte, the cation(s) and anion(s) of the compound are each well solvated by the water, and the solvated ions are not strongly associated with each other. In other words, they float around without necessarily staying close to each other. The result is that each ion can participate effectively in carrying charge - i.e. you have a strong electrolyte.

 

With a weak electrolyte, the cation(s) and anion(s) of the compound are not well solvated by the water, and so the cation(s) and anion(s) stay paired up (or worse, several compound molecules cluster together in solution). As such, the overall charge of the ion pairs or clusters are effectively zero, and so they cannot carry charge very well - i.e. you have a weak electrolyte. In the limiting case, the compound dissolves in water molecularly, meaning that neutral molecules are dissolved in water, rather than those molecules going on to dissociate into ions.