In a recent flight on an Airbus A380 I noticed that its ailerons are split into three segments which move independently.
What advantages does this design have? Do other aircraft have split ailerons (or other control surfaces)?
There are three main reasons for having multiple ailerons per wing on large aircraft:
On a large aircraft, at high speeds the material of the aileron can be flexed so much as to nullify or even reverse the direction of change of lift, which causes significant issues. The further out on the wing an aileron is, the more it is subject to these forces. For high-speed aircraft, this necessitates an outboard/inboard aileron, with the outboard being locked out at a certain speed limit. On the Airbus A380, the downwards motion of the outward aileron is locked out at 240 KIAS, and upward motion at 300 KIAS.
Another issue is that on most large aircraft, it is simply infeasible to have cables running through the aircraft, requiring the use of hydraulics. Having ailerons split would allow for more redundancy of system failures, allowing more controllability in these failure conditions.
Load Alleviation Function
Building on what Bret Copeland has already discussed in his answer, multiple ailerons can be used by fly-by-wire computers to flex the aircraft's wings in a specific manner to allow for less loading on the wing during cruise (counteracts the wing's natural tendency to flex upwards), as well as allowing for dynamic turbulence alleviation.
One of the examples of using this is in Airbus's Safety First magazine (July 2012), where they use the flexibility of having three ailerons to tune out lateral accelerations in the rear of the aircraft changing the gains of each of the controls. By being able to move the centre aileron a fraction after the inner aileron, they managed to avoid these initial shifts during flight testing of the A380.
The majority of airliners have multiple ailerons, however in some instances (such as the Airbus A310), there is no outboard aileron -- this function is replaced by spoilers.
Another example of using computers on multiple ailerons is the Boeing 747-8, with the use of the outboard aileron to fix issues relating to the development of flutter in certain extremes of the flight envelope, called the Outboard Aileron Modal Suppression (OAMS) system.
I don't know the specific rationale in this case, but if you think about mechanical advantage, the farther an aileron is out from the center of gravity, the more roll effect will have on the airplane. You'll notice in the image that the outer aileron is deflected less than the inner aileron, even though they are probably producing roughly the same roll moment.
A benefit I see is that it undoubtedly reduces wing-loading at the far end of the wings, and therefore reduces stress/flex on the wing during turns. This could be very important for an aircraft as massive and heavy as an A380, but less important for smaller aircraft where the wing-load is naturally less.