Winglets reduce "Induced Drag" which is created by all wings when they create lift. It is simply an indesireable, but inevitable by-product of lift.
Induced drag basically creates a rearward component of the lift vector at the wing tip, where the airflows from the upper and lower surface of the wing interact (they are forced to interact by tha abrupt ending of the wing!).
This rearward component acts like all rearward forces on an aircraft, as drag.
Now winglets are small vertical aerofoils which form part of the wingtip. They are usually specifically shaped and angled to the induced airflow to generate a small forward force (a negative drag, or "thrust"). They partly block the air from flowing from the bottom to the top surface of the wing, reducing the strength of the tip vortex which is the cause of induced drag.
Also, the small vortex caused by the winglet itself can be designed to interact with and further reduce the strength of the main tip vortex... clever stuff.
All in all this reduction in drag means less thrust means less fuel means less cost per-seat-mile and so on...
I know of no disadvantages of having winglets, though I recall a 747-400 (BA I think) which had the winglet bashed by something and it had to be removed. The aircraft continued to fly in revenue service for a week or so with only one winglet whilst a replacement was shipped over. The fuel penalty for having just one winglet was 4% or so.
There are pictures on the net somewhere of the "single winglet 747"!
I hope this helped you out.
I used to love the smell of Jet-A in the morning...