Most electrolytic capacitors are polarized, unlike other type of capacitors.
This means that the voltage applied to one terminal (the anode) should not be negative w.r.t. the other. So these capacitors cannot be used with AC signals without a DC polarizing bias.
The concept of using an electrolyte as one of the plates of a capacitor can be found in many patents of Inventor Charles Pollack.
Early Patents of Charles Pollack related to Electrolytic Capacitor were GB189601069 - Filed 1896-01-15, AT640 - Filed 1899-03-01, US672913 - Filed 1899-03-07
The earliest one GB189601069 can be considered as the patent that introduced the concept of an Electrolytic Capacitor, and hence, the basic or the Original patent of Electrolytic Capacitor.
Excerpt from Basic Patent: The invention is based upon the observation, which I have made, that if, in an electrolytic 'cell, the electrodes be such that, when the current passes through them, the plate, acting as a positive electrode for the time being, becomes covered with a film or coating, the effect of which is to insulate the metal of the plate from the electrolyte or, at all events, to offer a considerable resistance to the passing current, then a current of corresponding tension (say 100 volts) is precluded from passing through the cell altogether. And if the circuit be then broken, and the cell connected with a galvanometer, the impulse of the latter will correspond to the tension of the charging current. Hence it follows that the useful work or energy, of the current becomes stored up in the electrode. By the utilization of this fact I have produced an improved electrical condenser of great capacity
Patents from Charles Pollack introduced the technology however they were not usable designs but rather kind of experimental setups.
Major work on developing practical and applicable designs for Electrolytic Capacitor was done by Julius Edgar Lilienfeld.
Some of the earliest patents from Julius Edgar Lilienfeld related to Electrolytic Capacitor such as US1880263 and US1954636 filed 1930-06-19 and others describe innovations in design and manufacture of anodes, cathodes, separators etc of Electrolytic Condenser/Capacitor.