ESR is the sum of in-phase AC resistance. It includes resistance of the dielectric, plate material, electrolytic solution, and terminal leads at a particular frequency.
ESR acts like a resistor in series with a capacitor (thus the name Equivalent Series Resistance). 
The obvious question we need to ask ourselves is why would we want a “resistor” in series with a capacitor? The answer is that for bypass and storage purposes we don’t want it. (There are specific impedance compensating circuits called zobel networks where this is done on purpose, but that’s another story.)
In a perfect world our capacitors would have 0 ESR, but in the real world things don’t always work as easily as that. Therefore ESR is a fact of life and we need to find ways to work around it.
Very low ESR capacitors can be found but they normally come at a cost premium as compared to higher ESR capacitors. There are ways around this – using multiple capacitors in parallel will reduce the total ESR: because the ESR functions like a resistor, it follows the same rule as resistors in parallel. Parallel resistors have a total resistance lower than any individual resistor. Therefore parallel capacitors will have a lower ESR than any individual capacitor ESR.
This sounds easy enough, and in reality it works reasonably well. As with many things it does not work quite as well as it does on paper because we have parasitic elements in our circuit. Things like the capacitor leads and PCB tracks (or wire) have resistance and inductance which all have a part to play in diminishing the theoretical values. Don’t let this put you off from placing multiple capacitors in parallel – the benefits are real and definitely worthwhile. My intention is simply to inform you of some of the issues related to it and show why the textbook calculations may fall short in real life.
So what is the actual difference between using a low ESR capacitor and a high ESR capacitor?
To show you what happens comparatively I have done some spice based simulations.
The circuit is that of a 230V mains driven 300VA transformer with dual 25V AC secondaries. In order to attempt maintaining a reasonable approximation of the truth, I have included both the primary and secondary winding resistances into the otherwise perfect (text-book) transformer.
For the primary winding, the DCR is 4.7 ohm (R_Prim in the schematic below), and each of the secondary windings has a DCR of 0.1667 ohm (R_Sec1 and R_Sec2 in the schematic below). These values were taken from the datasheet of a well known and respected company making toroidal transformers.
Each of the secondary windings has its own bridge rectifier. In effect this acts as two independent power circuits – they do not interfere and / or influence each other. After the bridge rectifier we have a single 4’700uF capacitor C1, C2). In series with these capacitors you will see a resistance (ESR1, ESR2). These resistors approximate the ESR inside the capacitor in such a way that I can easily change the value to show you the effects. The values of these resistors do not necessarily represent real ESR values – they were chosen out of hand.
Lastly, each of the rectified and filtered supplies has a 1 Amp DC constant current sink (I1, I2) attached to simulate a constant load.
In the graph below we see the output voltage of the two circuits. The red graph represents the output voltage of circuit with the capacitor which has an ESR of 1 ohm (channel A). The blue graph shows the output voltage of the circuit with the capacitor which has an ESR of 0.1 ohm (channel B).
As can be seen from the graph above – the voltage ripple on Channel A is much higher in magnitude than the voltage ripple on Channel B.
For channel A (red), the voltage swings between around 29.8V at minimum to around 34.8V at maximum. That’s 5V P-P and around 3.5V RMS.
For channel B (blue), the voltage swings between around 32.6V at minimum to around 34.6V at maximum. That’s 2V P-P and around 1.4V RMS.
Just to have some kind of benchmark to work against, I include a simulation below of what happens if we have a perfect capacitor (with 0 ESR) and show it against the supply with the 1 ohm ESR capacitor below:
The green line (channel B) shows the output voltage of a power supply with 0 ESR (fictitious) as compared to the red line output voltage for a power supply with an ESR of 1 ohm (channel B). The total ripple voltage of the 0 ohm ESR capacitor is in the region of 0.08V P-P.
In this application (power supply filtering and storage), it seems that the lower the ESR of the capacitor used, the better voltage ripple on the output becomes.
The less the ripple, the better the “cleaner” the output voltage and less chance of ripple interfering with the circuitry that draws power from the supply.