Ion-propelled vs. Ion Thruster I am fifteen and an aspiring Mechanical Engineer. 
I read about ion-propelled aircraft, and was fascinated. On NASA's YouTube channels they outlined Ion Thrusters as a potential way to reach greater distance in space.
The way I see it Ion-propelled air-crafts are devices that use electrons to create air currents that help lift off the ground while Ion Thrusters actually accelerate ions to create currents in space that make traversing distances faster...? I am not sure if my understanding is correct and even it is could someone provide more details to further my understanding?
 A: Ion propulsion rockets are just that, rockets. They're just a different kind of rocket that the typical chemical propulsion rocket. Chemical propulsion uses chemical reactions and thermodynamics to create a high velocity exhaust, and hence thrust. Ion propulsion works by ionizing a gas (typically a gas with a high molecular mass such as xenon), accelerating the ionized gas to a very high velocity using electrical and/or magnetic field, and expelling that very high velocity gas from the spacecraft. Ion propulsion rockets tend to generate much higher escape velocities than the chemical rockets, but they also tend to generate significantly less thrust than do chemical rockets.
The upside of ion propulsion is their very high exhaust velocity, much higher than that from chemical propulsion. This means less propellant is needed than with chemical propulsion to achieve the desired goal. The rocket equation, which was mentioned in Floris' answer, is rather tyrannical. (Google "tyranny of the rocket equation" for more on this.)
As an example, suppose the mission at hand needs a delta V (change in velocity) of 12 kilometers per second. Using a chemical propellant with an exhaust velocity of 4 kilometers per second means the vehicle has to be 95% fuel at the onset. Most of the remaining 5% is fuel tanks to store all that fuel. Very, very little is available for payload. Using ion propulsion with an exhaust velocity of 120 kilometers per second means that only 9.5% of the initial mass is fuel.
One downside of ion thrusters is their very low thrust. This is okay if a lot of thrust is not needed. (It's not okay for launch, where an immense amount of thrust is needed. Ion thrusters currently just don't work as launch vehicles.) The Rosetta mission that is currently investigating Comet 67P/Churyumov-Gerasimenko and the Dawn mission that is about to investigate Ceres both used ion propulsion to get to their destinations. They needed lots of delta V, but not a whole lot of thrust.
One way to overcome the low thrust is to use a bank of ion thrusters. But that brings up their other downside, which is that ion thrusters consume a lot of electrical power to accelerate the ionized gas. This presently limits ion propulsion to use on smallish vehicles that have the luxury of time to get from Earth to the target. In other words, small robotic probes such as Rosetta and Dawn. Big vehicles such as the  Mars Science Laboratory (Curiosity rover) or something that could carry humans still need chemical propulsion.

Aside: The biggest beneficiary of ion propulsion is not interplanetary space probes. There just aren't very many interplanetary space probes to begin with, and many of them need high thrust.  The biggest beneficiary is commercial space. Space has been partly commercial ever since the 1962 launch of Telstar. There are several hundred birds in geosynchronous orbits, and a good proportion of these now use ion propulsion.
Getting a satellite into geosynchronous orbit is a very expensive proposition. A commercial communications satellite doesn't start turning a profit until after multiple years in orbit. The longer the satellite remains viable, the more profit. Communications satellites occasionally have to use thrusters to perform stationkeeping operations. Depleting all of its propellant spells the end of a communication satellite's useful life. (In fact, having a satellite deplete 100% of its propellant while still in a geosynchronous orbit is highly frowned upon. Operators of a geosynchronous satellite that is approaching the end of its useful life are supposed to make the satellite use the last little dregs of its fuel to boost the satellite out of its geostationary orbit into a satellite graveyard.)
Communications satellites that used chemical propulsion for stationkeeping only had a few years of profitability because of the low specific impulse of chemical propulsion. Ion thrusters enable communications satellites to turn a profit for a far longer period time. The vast majority of new communications satellites now use ion propulsion rather than chemical propulsion for stationkeeping.
A: Just a start... there are others on this site that know much more about these things (David Hammen, are you reading this?).
For any rocket to gain speed, it needs to expel some material - and the faster the material is expelled, the more speed the rocket gains. There is a "rocket equation" that shows the velocity you can reach depends directly on the exit velocity of the matter you expel and the ratio of initial and final mass:
$$v_f = v_{cx} \log\frac{m_i}{m_f}$$
Chemical (combustion) reactions limit the exit velocity $v_{cx}$ - so you need a LOT of fuel to burn (and throw out the back) in order to get up to speed. If you use an electric field to accelerate individual ions, they can go MUCH faster - if you accelerate them through 1 MV, the velocity of a proton being thrown out of the back of the rocket is almost relativistic - almost $1.4\cdot 10^7 m/s$ which is about 3000x faster than the velocity from a rocket engine (4000 m/s or so). This means you can reach a higher velocity.
These things work better at very high velocities in space - you probably don't want to use them on earth (I'm not sure what the interactions with the atmosphere will be if you start expelling a bunch of charged particles but I'm scared you will start a lightning storm).
A: This device is called the "Self Contained Ion Powered Aircraft". It is an ion propelled device with no moving parts, that lifts its power supply straight up off of the ground. See US Patent#10,119,527 Here is a link to a video of it and my website. The device works.
Video:   https://www.youtube.com/watch?v=Qdg0_hjuksQ&t=
Site:    electronairllc.org
