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Why is it necessary to appeal for an LBW dismissal when sometimes it is clear that the ball would have gone on to hit the stumps?
It is the duty of an umpire to give the decision as OUT when the ball running to the stumps, hits any part of the batsman other his bat. If the bowler or any of the player of the fielding side did not appeal for the dismissal, what can the umpire do in this situation?
Leg Before Wicket (LBW) is never as straight forward as other forms of dismissals like Caught or Bowled. More often than not, there is that element of doubt as to whether
The ball would have gone to hit the wicket or
The ball hit in line with the wicket or
The ball pitched outside leg stump or
If there was an inside edge before the ball hit the stumps.
The full rules to give the batsman out LBW can be found here.
The decision making for the above can neither be given to the batsman nor the bowler as their decision could be biased towards their own team. Therefore, the only neutral person who would be in a position to judge without bias would be the umpire, and this is why the bowler/fielding team needs to appeal to get an LBW decision.
The umpire, however, cannot give a batsman out LBW, even if were to be plumb, if there is no appeal from the fielding team for the dismissal.(Law)
Orangecrush's answer is technically correct, but relegates the most important point to the final paragraph: no batsman can ever be given out unless there is an appeal from a fielder. To quote from Law 27:
Neither umpire shall give a batsman out, even though he may be out under the Laws, unless appealed to by a fielder.
Hence it is necessary to appeal for all dismissals, whether they be bowled, LBW or timed out. (Note that Law 27 does not forbid a batsman walking).
Referring to the specific comments in your question:
It is the duty of an umpire to give the decision as OUT when the ball running to the stumps, hits any part of the batsman other his bat.
As noted above, this is incorrect. The umpire should not give a batsman out unless an appeal is made.
If the bowler or any of the player of the fielding side did not appeal for the dismissal, what can the umpire do in this situation?
The umpire should do nothing and wait for the fielding side to deliver the next ball of the over (or call "over" or "time" as appropriate if it was the last ball of the over/session).
Technically the fielding team should appeal for all dismissals, including bowled. The batsmen is well within his rights to rebuild his shattered wicket and continue his innings if the fielding team do not appeal, conversely if the fielding team WANT the batsmen to stay in, ie a strokeless blocker in a T20, and the batsmen deliberately drills the ball onto his own stumps or tamely straight at a fielder, the fielding team tell the umpire they are deliberately not appealing, but the batsmen walks anyway, he is out 'retired out' rather than bowled or caught!
