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I saw that some aircraft's rudder is composed of an upper rudder and a lower rudder (B727, A380, ...) as illustrated:

Flight control

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I wonder how does it work compare to a normal rudder and what are the pros and cons of the two-parts rudder over the one-part rudder?

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This is called Split Rudder and it provides redundancy. They run on different systems so if one fails, the other one can be used.

Here is a picture of a split rudder:

Split Rudder

Split rudders also provide a finer high speed control, in that only the lower one moves at high speed, reducing the exposed surface area and therefore the control effect.

Very importantly, there is a structural benefit. When we use the rudder it imposes a twisting load on the vertical fin. By only using the lower rudder, when the aircraft is at high speed, we reduce the twisting moment and transfer that load to a bigger, stronger, part of the airframe. The principle is the same as why the outboard ailerons are disabled at high speed.

The patent information is here.

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    $\begingroup$ To add a little context, in case a reader misses the significance... The rudder is the only control surface for which an aircraft only has one. Almost all aircraft have two sets of flaps, ailerons, elevators etc... Although it's possible to fly most aircraft without a rudder (or other control surface), it's not a comfortable experience and a split rudder makes this situation less likely. $\endgroup$
    – Jon Story
    Commented Oct 20, 2014 at 23:45
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    $\begingroup$ @JonStory In case of twin tail airplanes, there are two rudders. $\endgroup$
    – Farhan
    Commented Oct 20, 2014 at 23:55
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    $\begingroup$ Very true, I'll concede that point although I'll counter that there aren't too many twin tail aircraft around. Worth noting next time you're in a P-38 Lightning though :p $\endgroup$
    – Jon Story
    Commented Oct 20, 2014 at 23:56
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    $\begingroup$ Seems surprising that you can get a patent (filed in 1995!) for a split rudder design. Seems so basic and so obvious. Of course, the patent system is broken but still... $\endgroup$
    – RoboKaren
    Commented Apr 4, 2015 at 4:23
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    $\begingroup$ I've been a huge fan of planes since I was about 9 years old, , 13 years later and until now I had never ever ever heard of or seen a split rudder before! $\endgroup$
    – Kxy
    Commented Aug 11, 2017 at 14:57
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As pointed out by Farhan, it adds redundancy to the aircraft.

A 747 once experienced a rudder hard over. Even with the split rudder design, the crew had to rely on asymmetrical thrust to control the aircraft. If it had one very large rudder surface, control would be much more difficult, or may even be impossible.

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It allows one part to jam while the pilot retains control over the other part.

Also the higher portion of the rudder induces roll when deflected, at high speeds (where control surface deflection is more effective) only the lower portion can be deflected to minimize this effect.

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