I can't answer if the electrodes would or would not become charged without completing a circuit.
Voltage however is the difference in electric potential. In the case of a single cell battery the voltage is determined by the difference in the reduction potential between the cathode and the anode. Whichever one has a higher reduction potential (the cathode) will draw electrons from the anode.
Current on the other hand is a measure of "how much" electricity is actually flowing through the circuit (i.e the rate of flow of electrons).
Voltage and current are connected. Think of current as the actual particles and the voltage as pressure behind them. Much like a water system, if you open a tap, water will flow. But if you add more pressure the water tank then the water will flow faster and go further.
If you have a huge amount of electricity but no voltage to push it it will not be able to flow through a circuit, hence you cannot use batteries of smaller voltage to run a larger circuit board (for example) than it's designed for, the low voltage won't be able to push the current through it (i.e the resistance of the circuit is too high).
Now pure water is not actually a very good conductor of electricity, it offers a high level of resistance to electrical currents. A small battery will not be able to drive electricity far through water. It might be able to form a circuit through a very very small distance. Electrolytes such as the salts you were using increase the conductivity (decrease the resistance) of the water by introducing ions into the solution making it easier for electricity to flow through it with a lower voltage. Acids and bases can also be used as electrolytes to increase the conductivity of water for as long as they form ions in solution.
I'm not 100% sure how this works but if I recall correctly the anode donates electrons to the ions whereas the cathode takes electrons from the ions. This is what causes the negative and positive clouds around the electrodes. The electrolyte pretty much carries electrons from the anode to the cathode.
My apologies if I at any point confused cathode and anode. Remember that electricity is the flow of negatively charged particles along a charge gradient.
Hope this helps!