What is a stabilized approach?
When you look at FAA publications the criteria for a "stabilized approach" is pretty simple:


  A stabilized approach is one in which the pilot establishes and maintains a constant angle glidepath towards a predetermined point on the landing runway.




When pilots talk about stabilized approaches we generally mean a little more than that though - for example the SKYbrary description of a stabilized approach, the Flight Safety Foundation, and Airbus all include more than just the constant-angle glide path in their recommendations for what makes an approach "stabilized".

Criteria that are commonly included in a "stabilized approach" are:


Maintaining a constant-angle glidepath toward a predetermined aiming point on the runway.
Maintaining a specified descent rate
Maintaining a specified airspeed (Vapp)
(Generally being slightly above Vapp is OK, but below is unacceptable.)
Having the aircraft configured for landing (gear, flaps, etc.)
All required checklists completed
The approach can be maintained with only "small corrections" to pitch, heading, or power.


An approach is "stabilized" if all the required criteria are met by a specific altitude (Airbus calls them "Stabilization Heights", and recommends 500 feet for approaches in VMC, and 1000 feet for approaches in IMC).
If the approach is not stabilized by the appropriate stabilization height (or conditions cause the approach to become "destabilized" - a sudden change in winds for example) the approach is abandoned (a "go-around").



So what makes stabilized approaches safer?
The theory is that a stabilized approach reduces the pilot's workload as they're getting closer to the ground: If the aircraft is configured for landing and all the checklists have been completed all the pilot has to do is gently nudge the controls to keep the plane going where they want it, and they can concentrate all of their attention on a nice, safe, (hopefully) smooth landing.

Are stabilized approaches always safer?
From a workload perspective, yes. It's hard to argue with "The pilot has less work to do".
From a "What if my one-and-only engine dies?" perspective, maybe not.  

Stabilized approaches generally require partial power to be maintained throughout the approach, and although the engines in modern single-engine piston/jet aircraft have a pretty good track record of reliability there are good arguments for "non-stabilized" approaches (or approaches that are effectively "stabilized" at idle power) in single-engine aircraft.  

Even if you're not flying a "stabilized approach" the way the airline folks define it you're well-served by incorporating some of the elements though - going around if your checklists aren't completed by a certain altitude, abandoning the approach if you have to manhandle the controls to try to keep things together, etc.