You can split water by electrolysizing it with an electric current. So in other words, pass an electric current through water, and you will break it apart into hydrogen and oxygen gas.
But there is a catch, water does not conduct electricity. Pure water will not conduct, but if you add electrolytes it will. I assume that the "ordinary" water LeonhardEuler referred to is tap water, which does contain additives (like fluorides), and some of these additives will break into ions when dissolved in solution and allow electric current to pass through the water.
In electrolysis, just about everything in/around/making up the "Cell" as it is referred to matters.
The electrolyte matters, some electrolytes will break into ions more readily than others. Some good electrolytes are Sodium Chloride (NaCl), and Sodium/Potassium Hydroxide (NaOH or KOH), these will readily form ions in solution.
The electrodes matter, a lot. Certain electrodes are more suitable to be used as anodes (+) or cathodes (-) than others. For instance, you do NOT want to use iron/steel as an anode because it will be oxidized in the reaction, and turn into rust and dirty up you water (and stink!). It is much better to use something like graphite as the anode (or more ideally platinum $$$$$), since it won't break down so easily. Mind you, even graphite will erode away with time. For the cathode, you have much more freedom of choice since it is protected by a negative pole, but I myself still use carbon since it is cheap and available to me. Even the placement of the electrodes matter, the more distance you put between them, the more resistance there is in the cell, and since Current goes down when resistance goes up (V = RI), you won't be making as much gas with less current.
There is a minimum voltage needed to electrolysize water, I think it is somewhere around 1.2 volts, but I am too lazy to look up the exact number. But keep in mind, the production of the gasses is dependent on the current through the cell, not the voltage.
Also, if you use NaCl as you electrolyte, you technically won’t be making H2 and O2, you'll get H2 and Cl2, since it is easier to reduce chloride to make chlorine gas than it is to reduce oxygen to make oxygen gas. So in the long run you will be turning your NaCl into NaClO3 and make a lot of hydrogen and chlorine gas.
As you might be able to tell, I have a passion for electrolysis and I have lots of first hand experience with this.