If you want to avoid Wikipedia, look at the following NASA resources:
Overview of Hall Effect Thrusters
Ideal Rocket Equation
The NASA overview is short and easy to read, so there's no need to re-explain how the thruster works. To expand on the discussion in this question, there are (at least) two circumstances where a Hall thruster is a much better choice than a chemical rocket.
When you need high precision stationkeeping: if you want to make precise adjustments to the orientation of a satellite, it helps to be able to control thrust in very small increments. This is a big challenge for chemical rockets, but Hall thrusters can easily adjust their thrust in milliNewtons by changing discharge voltage and/or propellant mass flow rate.
When you need a high ratio of thrust produced to propellant used: Ion thrusters will usually outperform Hall thrusters here, but compared to chemical rockets the Hall thruster is still a much better choice. Read about Specific Impulse to lean more.
As you probably know, the weight of components is immensely important for space launch, so scientists and engineers make significant efforts to minimize weight wherever possible. For propulsion, if we can make efficient use of our propellant, then we don't need to carry as much of it, thereby reducing weight. The second link above shows the ideal rocket equation
$\Delta{u}=v_{eq}ln(MR)$
where $\Delta{u}$ is the change in the spacecraft's velocity, $MR$ is the ratio of empty spacecraft mass (no propellant) to full mass (full of propellant), and $v_{eq}$ is related to the exhaust velocity of the propellant. So, suppose you have a propulsion system and you plan to use all of your propellant to get the biggest increase in velocity possible: you want to maximize $\Delta{u}$ for a fixed $MR$. As you can see from the equation, to maximize $\Delta{u}$ you must make the exit velocity $v_{eq}$ as large as possible.
This is where the differences between propulsion systems become very clear. You can perform additional research as needed, but its not hard to find sources for the following typical exhaust velocities:
Liquid fueled chemical rocket: under $5000$ $m/s$
Hall thruster: around $20000$ $m/s$
Ion thruster: around $50000$ $m/s$
When people tell you that Hall thrusters are "more powerful" than chemical rockets, they really mean that Hall thrusters give you more thrust for a given amount of propellant mass. In terms of thrust alone, chemical rockets win: There is no electric propulsion device in existence that can launch a rocket from earth's surface into orbit. Chemical rockets create high thrust by moving an enormous amount of propellant mass. Another good parameter for additional plasma thruster research is Thrust-to-power ratio ($T/P$).
