The difference between these and your standard computer speakers can most easily be seen by hearing a song with a high bass output on standard speakers versus out of the monitors.
As you begin your search, you may run into the terms Far-Field and Near-Field.
Far-Field monitors are intended to be more than ~6 feet away from the listener.
These are meant for more of a pro-style acoustically treated studio setup. Or in other words... not for us!
Near-Field monitors are what you should focus on.
They are designed to be placed less than 6 feet away from the listener.
Since they are positioned close to you, they are relatively unaltered by the room acoustics.
This is good news for the home studio dweller as our rooms are certainly less than pro-quality.
So now we've established that our focus will be on near-field monitors, let's talk about the two flavors: Passive and Active.